A WORKBOOK FOR
Arithmetic We Need
o
BROWNELL
SAUBLE
CURR HISTJ
QA
106
B98
A
gr . 6
wkbk .
' c . 2
0JC IIBB15
Large Numbers in the News
A
University Gifts
Reported
The local University reports a
total of $1,278,952 received in
gifts during the past year.
B
Advt. FIRE SALE! Advt.
TV Sets
Entire stock of TV sets, valued
at $80,000, to be sold at great
reductions. From burned ware-
house, but sets not damaged.
c
Stamp Auction Well
Attended
Largest single sale at the Stamp
Collector’s Auction was the fa-
mous Brown collection of more
than 367,000 stamps. _____
D
Order Your
SATURDAY COURIER Now
Last week’s issue (42,000) of
the Saturday Courier was not
enough to fill the demand. Order
your copy saved for you.
E
United Charities
Appeal
This year’s goal is set at
$1,230,000. Be ready to give gen-
erously in this good cause.
F
APPLE BLOSSOM
TIME AGAIN
An estimated 750,000 people
will drive through our lovely
Blossom Valley to see the apple
trees in their first bloom.
Miss Day asked the class to look for large
numbers in newspapers and magazines. The
clippings shown above are some of the ones
which the class found.
After each pupil had cut out at least two
clippings, Miss Day mixed up the clippings.
She asked each pupil to draw two and explain
the meaning of the numbers in them.
1. Write in words each number in the clippings above.
A. $1,278,952
B. $80,000
C. 367,000
D. 42,000
E. $1,230,000
F. 750,000
2. For $1,278,952, write the figure that is in
a. one’s place:
c. million’s place:
e. hundred-thousand’s place: _ .
— -
b. ten’s place:
d. hundred’s place:
f. ten- thousand’s place:
4 5 7.4
The Meanings of Numbers
Complete the following statements:
1. The 1 -place number 9 means ones.
2. If we add 20 to 9, the sum is
3. In 20 there are ___ tens, so the number
29 means tens and ones.
4. If we add 500 to 29, we have
5. From Ex. 1-4, we see that the number
529 is made up of
hundreds and tens and ones.
6. The number 27,529 is made up of 27
and 5
and 2 and 9
We read 27,529 as “twenty-seven thousand
five twenty- ”
7. Another way to explain the number 27,529
is to show that it is made up of five products.
Complete the table below.
27,529 is made up of
2 X 10,000 = 2 0,000
7 X 15000 =
5 X =
X =
9 X -
The sum is
8. Copy the numbers in the box below with
the figures in the proper places to show what
each figure means.
9. There are many ways of taking a number
apart to show what it means. Complete the
statements below, which tell the meaning of the
number 27,529 in five different ways.
27,529 means
thousands and hundreds and
tens and ones,
or hundreds and tens and
ones,
or hundreds and ones,
or tens and ones,
or ones.
10. Explain the meaning of the figure 4 in a-c below.
a. In 3,543, the 4 means 4 , or 40.
b. In 3,504,319, the 4 means 4 ,or
c. In 4,100,325, the 4 means 4 ,or
UHISffiRSITT
Reading and Writing Larger Numbers
•a
a
.2
(3
(8
X
Value of
Place ->■
!j
1 §
•a £
o id
2 e
£ a
I. » X)
’O 3 q
<U O 3
•d
<u
u •- a
"d £> O
£ s|
■ss I
u
•d
c a a
Sco
c a u
3 gs
X H ffl
I |
3 §a
33 H <
1 |
c a> X
33 H H
1 1
3 S s
33 H O
1
Value of
1
Period ->
Billion
Million
Thousand
One
a.
9 2 ,
8 9 7 ,
4 1 6
b.
2 ,
3 9 8 ,
0 0 0 ,
0 0 0
c.
1. Number a, above, tells the average dis-
tance in miles of the earth from the sun during
its rotation. Write the words you say when
you read 92,897,416.
[ Through billion’s period ]
c. One billion one hundred twenty-six million
one hundred thousand two hundred nineteen.
5. In a recent year, United States airlines
flew 15,548,247,000 passenger miles. Write this
number in the box, beside c.
If you are not sure where to start writing
this number, say the periods, beginning with
one’s period, as follows:
000 means ones ; 247 means ;
548 means ; so 15 is
in period. The 5
should be written in billion’s place, and the 1
in place.
6. Look at Ex. 4 again. Underline the words
that name the periods.
Which period is not named by the printed
words?
2. In 92,897,416 there are figures, each
with its own place value, and
f
periods. The periods are _ OTZ&k ,
, and
3. Number b in the box above is read:
_
4. Write the following numbers in figures:
a. Fifty-six thousand two hundred sixty.
b. One hundred forty million six hundred
twenty thousand three hundred three.
7. 10 = 10 X 100 = 10 X ;
1,000 = 10 X ; 10,000 = 10 X
As we move to the left in a number, the value
of each place is times the value of the
previous place.
8. Put commas in the following numbers to
show the periods:
a. 82653147 c. 7065320
b. 139428 d. 51074931
9. Copy the number in Ex. 8 that has
a. 9 in hundred’s place.
b. 653 in thousand’s period.
c. 5 in thousand’s place.
d. 13 ten thousands.
3
Round Numbers
If there are 194 pupils in Grade 6, we can
say that in round numbers there are about 200
pupils.
Number line A, below, will remind you about
rounding numbers.
1 1
CO -3*
R m
i
- 132
- 133
CO
1
in m3 t'. oo o c
co co co co co ^
1 1 1 l i
' ’ ’ i | ' ' '
Round to 130
1
1
Round to 140
1. Rounded to the nearest ten, 131 is called
, because 131 is nearer to
than to
2. 135 is exactly halfway between 130 and
140. By the rule we follow, 135 is rounded
to the next larger ten; so 135 is rounded to
Round each of the numbers below to the
nearest ten.
3. 132
6.
128
4. 129
-- 7.
144
5. 143
___ 8.
127
9. Number line B, below, shows the whole
numbers from 110, or tens, to ,
or tens.
10. Study number line B. Which numbers
between 110 and 120 would be given as 110,
when rounded to the nearest ten?
11. On number line B, circle each number
from 110 to 130 which, to the nearest ten, would
be rounded to 120.
12. When you round 836 to the nearest
hundred, you think , “836 is nearer to
hundred than to hundred, so 836 is
rounded to ” You notice especially
the ’s figure, 3; you do not have to
notice the one’s figure.
13. Round to the nearest hundred:
683 7,261
429 35,738
14. Round to the nearest thousand:
2,199 79,625
8,507 132,479
15. Round to the nearest million:
31,275,859
8,076,742,108
You have to use good sense about when to
round a number and how far to round it.
16. Suppose your train leaves at 19 minutes
before 3 o’clock and you write the time in
round numbers as 20 minutes of 3. Is this
a good way to round 19?
If the train leaves at 22 minutes of 3, is it
a good idea to round the time to 20 minutes
of 3?
17. If you need exactly 73 inches of molding
for a picture frame, and you round 73 to the
nearest 10, how many inches will you get?
Is this a good way to round 73?
B -l — h
A 1— I *-
-I— I J-
-I 1 1 1 1 1 1 1 1 h
A 1 1 1 1 1-
4
Roman Numerals
Roman number symbols: I V X L C D M
Arabic number symbols: 1 5 10 50 100 500 1,000
1. On this clock, the hour hand has passed
nine; the minute hand points to The
time is minutes past o’clock.
Ex. 4-9 show how to think with the six
subtracting pairs of letters.
4. IV = V - I, or 5 - 1, or
5. IX = X — I, or 10 — 1, or
6. XL = L - X, or 50 - 10, or
7. XC = C - X, or 100 - , or
8. CD = D — C, or 500 — ,or
9. CM = M — C, or 1,000 — ,or
10. Write with Roman numerals:
a b
2. Write the hours in Arabic numerals around
the edge of the clock.
Writing numbers with Roman numerals is
something like making change. You take a
number apart and then use the Roman number
symbols like coins to build up the right
amount.
3. Show why 68 is written as LXVIII.
10 =
20 =
30 =
40 =
50 =
60 =
Think , “68 = 50 + 10 +5+1+1+ 1”;
70 =
100 =
200 =
300 =
400 =
500 =
600 =
700 =
write, “LX
80 =
800 =
To check, look at the Roman numerals that
90 =
900 =
you wrote and count “50, 60,
100 =
1,000 =
jj
In Ex. 3, the values of all the letters used
are added from left to right.
Besides the single letters, there are six sub-
tracting pairs of letters in the Roman system
of notation. In using these subtracting pairs
to help us build up larger numbers, we think
of each pair as a single number and add its
value to the values of the other letters.
4 =
9 =
40 =
90 =
400 =
900 =
6 =
11 =
60 =
110 =
600 =
1,100 =
5
More about Roman Notation
1. The seven letters used in the Roman sys-
tem of notation are as follows:
I = ____ V = ____ X = ____ L = ____
C - D = M =
2. Remember the way to write and to read
Roman numbers! In general, values of the
letters are
from left to right; and when one of the sub-
tracting pairs is used, we think of it as a
single number and its value to the
other values.
3. Tom wrote 49 as XXXXVIIII. Show
how Tom took 49 apart.
In Roman notation, we should use the small-
est possible number of letters.
Think , “49 = 40 + In Roman nota-
tion, 49 = ”
4. Jed, Marge, and Sally read XXIX in three
ways. Cross out the wrong ways.
Jed: XXIX = X+ X+ I+ X=31
Marge: XXIX = X +XI +X = 10 ■ +11 + 10 = 31
Sally: XXIX = X + X + IX = 10+ 10 + 9 = 29
5. Write in Roman notation:
84
192
516
406
709
969
1,251
614 __
6. Write three Roman numbers, using the
three letters I, X, L in each. Then write them
in Arabic notation.
7. The last chapter in a book was numbered
XXVIII. There were chapters in the
book.
1. Draw a dotted line in this Roman nu-
meral to make half of ten become
five.
[ Using Roman numerals]
5. What number is one greater when one is
taken away from it?
2. Can you make it look as if
half of nine is four?
6. Take one hundred from
four hundred and leave five
hundred.
3. Make half of twelve become
seven.
4. What number becomes ten
more when ten is taken away
from it?
7. Write SIX and take away nine. On the
same line, write nine and take away ten. Then
write forty and take away fifty. Show that
you have SIX left!
6
A Review of Addition
1. Sue has saved 4 box tops, and Ann has
saved 3. Together, they have box tops.
Adding is quicker and easier than counting.
The work on this page and the following pages
will help you to find out how well you remem-
ber the things that you have learned about
addition.
2. In the addition 7 + 29 = 36, the num-
bers 7 and 29 are called the ;
36 is the
3. Only things that are in
some way can be added. In order to add 5 car-
nations and 9 roses, we must think of them
all as
4.4+7=11 is an addition fact. If you
know the fact 4+7 =11, you also know
its reverse, 7+ =
5. In the space below, draw a dot picture
to show how you would add 8 and 6 by first
making a ten.
The sum is
[Addition facts ]
First
say the
answers.
Then write the
sums.
a
b
c
d
e
f
6.
2
0
4
9
6
1
+ 2
+ 1
+ 5
+ 0
+ 5
+ 8
7.
1
4
0
5
8
6
+ 1
+_7
+ 0
+ 8
+ 3
+ 9
8.
3
9
1
6
4
4
+ 5
+ 4
+ 0
+ 7
+ 8
+ 2
9.
9
2
6
1
5
0
+ 7
+ 0
+ 4
+ 2
+ 4
±2
10.
8
7
3
1
2
3
+ 6
+ 7
+ 8
+ 9
+ 4
+ 0
11.
7
8
7
5
8
3
+ 5
+ 8
+ 9
+ o
+i
±3
12.
5
4
8
6
5
4
+_!
+ 6
+ 5
+ 6
+ 9
+ o
13.
2
8
1
0
3
5
+ 9
+ 7
+ 3
+ 5
+ 6
+ 5
14.
1
8
7
1
9
0
+ 7
+ 2
+ 6
+ 5
+ 9
+ 7
15.
6
7
2
0
7
6
+ 3
+_!
+ 6
+ 3
+ 4
±8
16.
2
9
0
3
2
4
+ 3
+ 1
+ 4
+ 7
+ 5
+ _1
17.
6
8
3
2
1
7
+ 0
+ 9
+ 2
+ 1
+ 6
+ 2
7
Addition Families
The facts below are all part of the 3+4
family. If you know 3+4 = 7, you can
quickly write the other facts in this family.
Finish the additions below.
1. 3 13 3 53 83 3
+ 4 +4 +34 +4 +4 +64
7 17 37
2. Write four other facts in the 3+4 fam-
ily. Be sure to include the answers.
Write the numbers that will make the follow-
ing belong to the 3+4 family:
3. 3 3 6 3
+ + +---- +
57 47 67 77
4. In each fact in the 3+4 family,
a. the figure in one’s place in the sum is
; b. the ten’s figure in the sum is the
same as the figure in the larger
addend, because the sum of the figures in one’s
column is less than
Finish these facts in the 8+6 family:
8
1 8
8
58
48
+ 6
+ 6
+ 36
+ 6
+ 6
1 4
24
6. 8+6 =
or ___
ten and _
__ ones;
so in the 8+6 family you always have _ _ _
ten to carry to the column.
This makes the ten’s figure in the sum ___
more than the ten’s figure in the larger
addend.
8
[ Higher-decade A.; without and with carrying ]
The facts in these addition families are the
kind you must use often in column addition
and in multiplying. Here is a chance to get
practice that will help you.
a b c
7. 2 5 9 4 2
+37 +5 +4
d e
4 68
+ 19 +9
8. 4 5 9 1 7
+7 +23 +2
28 3 8
+ 8 +5
9. 9
+ 14
27 73 3 36
+5 +6 +57 +3
10. 5 7
+ 6
42 35
+ 5 +9
26 85
+ 8 +2
11. 3 7
+ 9
56 69
+ 4 +9
44 73
+ 8 +8
A ftunb&i
and Pulled.
1. In adding, would you say, "6 and 7 is 11”
or "6 and 7 are 11”?
2. This is a very old rhyme:
Every lady in the land
Has 20 nails upon each hand
Five and twenty on hands and feet
And this is true without deceit.
See if you can punctuate this rhyme so it
can be read sensibly.
Column Addition
1. The same addends are used in each of
Ex. a-d below, but they are added in a dif-
ferent order each time. What is true of the
sums?
[ Order in adding; 1 -place addends ]
3. To make sure that the sum in the box
is right, check by adding upward.
Think , “5, , ” Is the sum right?
5
3
4
+ 9
21
b. 9
4
5
+ 3
21
3
4
9
+ 5
21
d. 4
3
5
+ 9
21
We may add groups or numbers in
any order.
2. In the box, adding
downward, look at 3 and 2
and think , “ ” Re-
membering this sum, 5,
look at 1 and think , “ ”
The last two numbers to be added are
and Write the final sum in the box.
Add downward in rows 4-6. Check by add-
ing upward.
a b c d e f
4. 2 3 5
4 1 2
+5 +4 + 7
5. 4 2 3
2 5 0
2 0 4
+5 +4 +5
6. 4 5 4
1 2 0
4 1 5
+3 +4 +4
8 2 3
1 7 3
+6 +1 +4
6 4 2
2 2 0
1 0 3
+ 6 +7 +6
7 3 3
0 4 5
2 2 0
+7 +3 +4
a b c 1. Find these sums:
Column a:
Column b:
Column c:
Row d: Row e: Row f:
Diagonal from a: 4 + 5 + 6 =
Diagonal from c: 2 + 5 + 8 =
What do you notice?
2. Write the numbers used in the square in
order, beginning with the smallest.
, — , — , — , — , — , — , — ,
What do you notice about these numbers?
This "magic square” was discovered many
years ago in China. Both the square and the
number 15 were used as good -luck charms in
Eastern countries.
9
1. To add the numbers
in box A, we can separate
them into their parts as in
box B and then add the
parts.
Finish the adding that is
started in box B. Then
put the sums together to
find the final sum. Write this sum in box B.
Adding Larger Numbers
[With carrying ]
Add downward and write the sums. Then
B
4,000 and
5 0 0 and
8 0 and
1
7 0 0 and
9 0 and
6
+ 1,000 and
8 0 0 and
3 0 and
5
5.+Q0Q and _
_ _ and
and
---
2. Now find the
sum in box A.
3. Complete the following sentences to show
how you added in box A:
a. You find the sum in
column first. The sum of the ones is ,
so you write “ ” in one’s column in the
sum and carry ten to ten’s column.
b. When you find the sum of ten’s column,
you write “ ” and carry hundreds to
column.
c. In hundred’s column, you think , “7, ,
” You write “ ” and carry
thousands.
d. There are thousands in all.
In adding, write ones in the sum under
ones, tens under tens, and so on. Carry
1 or more to the next column when the
sum is 10 or more.
check by adding upward. Write your check
sums on the dashed lines above the examples.
4. 3,182
654
+ 97
753
985
+ 585
6,895
978
+ 2,837
Here is a short review of some of the number
facts you use very often in column addition.
Write the sums quickly.
a
b
c d
e f
5. 5
1
2 5
8 9
+ 7
+ 8
+ 6 +5
+
1
1+
1 00
6. 9
4
6 7
5 9
+ 3
+ 7
+ 8 +6
+ 8 +9
Remember: Addition families help you in
column addition.
Write the sums.
a
b
c
d e
7. 2 4
3
45
8 7
+ 8
+ 18
+ 9
+ 32 +19
8. 3 2
6 1
4
28 30
+ 9
+ 5
+ 29
+ 8 +7
Add
downward
in row 9.
Add upward to
check,
and write the check sum on the dashed
line above each example.
a
b
c
9. 3,
063
694
9,286
124
25 1
34 1
7,
36 1
332
9 1 4
+ 550
+ 873
+ 2,620
70
Meanings in Subtraction
[. Remainder ; difference]
These stories about Queenie’s pups show some ways of using sub-
traction. Write the subtraction fact for each problem. Use the dot
picture to help you. Then answer the question below the problem.
1. 4 of the 7 pups were black, and the rest
were tan. How many pups were tan?
0000m© #
If 4 is one part of 7, the other part is
2. If Bill wants to keep 2 pups, how many
of the 7 pups can he give away?
If 2 out of 7 pups are left, how many are
gone ?
4. The biggest pup weighed 8 ounces at
birth, and the smallest weighed 3 ounces.
a. What was the difference between their
weights?
WWi®
® ® m ®
b. The biggest pup was how many ounces
heavier than the smallest one?
3. After Bill gave away 3 of the 7 pups, how
many pups were left?
000m 0 © •
If 3 pups are taken from 7, how many are
left ?
8 are how many more than 3?
c. The smallest pup weighed how many
ounces less than the biggest one?
3 are how many fewer than 8?
In subtraction, we know the size of one of two groups or
numbers and the sum of the two groups or numbers. We sub-
tract to find how many are in the other group or number.
5. In Ex. 1-3, the other number is a re-
mainder. It tells the other part in Ex. 1; the
number gone in Ex. ; the number left in
Ex
6. In Ex. 4 the other number is the differ-
ence. In Ex. 4b, the difference tells how many
; in Ex. 4c, it tells how many
11
A Review of Subtraction
C
X
1. From the drawing above you can see that
4+5 = and that 5+ =
2. By covering dots you can prove that
9 — 4 = and that 9 — 5 =
3. The whole story in A. and S. about 4, 5,
and 9 has these four related facts:
Finish the whole stories in A. and S. that
are started in Ex, 4-7.
4. 7 + 5 =
5. 9 +
= 13
6. 8 + 9 =
7. 15 = 9 + 6
Addition facts help us to remember
subtraction facts.
8. To help you with 13—8, you may think ,
“13 = 8 + then you also know that
13 - 8 =
Give the A. fact that helps in Ex. 9-11.
9. 17 -9 = ____ 17 =9 + ____
10. 11 -4 = ____ 11 =4 +____
11. 15 - 6 =
12
[Subtraction facts ]
If you can write these facts quickly, you
can subtract any numbers:
a
b
c
d
e
f
12. 8
10
15
0
9
11
- 5
- 5
^6
- 0
- 5
- 4
13. 15
8
5
9
16
3
-7
- 2
- 1
- 9
- 9
- 0
14. 11
17
10
12
7
12
-8
- 9
-3
- 3
-4
- 6
15. 16
11
4
8
12
10
- 7
- 7
-0
- 3
-9
- 4
16. 14
7
9
13
1
15
- 7
-2
- 6
- 5
- 0
- 8
17. 14
8
9
13
7
15
-5
- 1
- 3
-8
- 7
- 9
18. 8
6
11
13
6
16
-0
-3
- 2
- 7
- 1
X*
19. 6
9
10
2
11
14
- 2
- 1
- 6
- 0
-3
- 9
20. 9
14
12
9
7
12
-2
- 6
- 7
-0
- 1
- 5
21. 17
13
1
18
8
7
-8
- 6
- 1
- 9
-4
- 3
22. 8
10
10
13
11
14
- 8
- 1
- 7
- 9
- 5
- 8
23. 12
11
7
9
12
6
-4
- 6
-0
- 4
- 8
- 6
Each subtraction fact is part of a family, just
as each addition fact is. If you know the sub-
traction fact, you can easily say remainders for
other examples in that family.
Subtracting with Two-Place Numbers
[Without and with borrowing ]
You can subtract 2-place numbers very quickly
too. Study boxes C and D.
1. Write the answers missing in box A.
2. Complete the rest of the 7—4 family:
37-4 = 77-4 = ___
67-4 = 97-4 = ___
g
Some members of the 1 3
— 9 family
1 3 23 33
63 93
- 9 - 9 - 9
- 9 - 9
3. In box B, the ten’s figure in each answer
will be 1 than the ten’s figure in
the minuend.
4. Write the answers in box B.
Complete each fact below and tell its family.
5. 47 — 6 = ( family)
6. 30—8= ( family)
7. 29— 5= ( family)
8. 64—9= ( family)
Write the remainders for Ex. 9-14.
9.
25
- 9 =
12. 42
- 6 =
10.
60
-4 =
13. 54
-3 =
11.
76
-9=
14. 83
-6 =
C
9
39
89 = 80 and 9
- 5
-5
— 45 = 40 and 5
4
34
40 and 4, or 44
D
13
23
53 = 40 and 13
-8
- 8
— 28 = 20 and 8
5
15
20 and 5, or 25
15. Subtract in box E. Think , “73 = 6 tens
and ones.” Then think ,
“13 - 8 = and 6 - 2 =
” 73 - 28 =
Subtract in rows 16-20. Try to work with-
out showing the borrowing.
16. 5 6
- 24
b
82
-30
c
35
- 3 1
17.
39
32
17 63
16 - 48
18. 4 1
- 26
19. 3 1
- 1 0
20. 4 5
-34
39
23
72
1 6
96
48
84
57
56
27
83
77
d
64
- 1 9
79
-76
29
24
49
4 1
22
1 2
e
78
-48
85
69
6 1
47
72
55
34
1 8
73
Subtracting Large Numbers
[8- and 4-place minuends ]
1. Bill made _3_ hits out of ___ shots. He
missed ___ times. The S. fact is
Write the remainders for Ex. 2-16.
a
b
c d
e
f
2. 5
7
9 8
6
7
- 3
-4
- 7 - 5
-4
- 5
3. 9
5
7 8
8
9
- 5
- 2
- 3 - 6
- 2
- 3
4. 7
9
8 9
8
9
- 2
- 6
- 3 - 2
- 7
- 4
5. 11 -
2 =
10. 12
- 9 =
6. 13 -
6 - ___
ii. ii
- 5 = _
7. 16 -
7 =
12. 15
- 7 =
8. 17 -
9 =
13. 14
- 9 =
9. 15 -
8 =
14. 12
- 4 =
a
b
c
d
e
15. 11
13
16
13
12
- 6
- 7
- 8
- 9
- 3
16. 15
11
13
14
11
- 6
- 4
-5
- 7
- 8
74
3@
64^
- 28
6 1 5
2®
4
164
190
5©@
2,$ $ /
1,5 8 3
1,0 9 8
tell
- 3,7 8 5
2,9 4 6
The examples in the
about borrowing.
box will
remind you
Write the remainders for rows 17-25
below. Try to think the borrowing.
a
17. 5 6 4
- 83
b
338
- 73
c
678
-475
d
6,347
-3,524
18. 8 3 4
-366
674
-579
925
- 832
7,762
- 2,364
19. 4 7 3
-394
326
- 1 76
647
-455
6,814
-4,207
20. 9 2 9
- 698
4 1 9
- 367
38 1
- 1 95
1,629
- 982
21. 5 2 4
-478
84 1
-789
923
- 429
2,425
- 548
22. 7 8 8
-289
984
-546
95 1
- 338
8,5 19
- 3,775
23. 8 6 7
-452
783
-392
934
- 544
5,577
-2,868
24. 4 3 1
- 285
567
- 342
246
- 1 57
9,6 14
-8,754
25. 5 9 6
-489
472
-375
375
- 28 1
4,475
- 2,525
Zeros in Subtraction
2 9 (14)
4 9 <W)
3 9 (§)(§)
4 9 9,(13)
-5D0
A&00
-5&tr2
-65
- 354
- 1,5 9 7
- 325
239
146
2,4 6 3
4,6 7 8
The examples in the box show how you
borrow when there are zeros in the number
you subtract from. Here is some special prac-
tice that will help you.
a
b
1.
10-1 =
9 - 3 =
2.
9 - 0 =
10 - 2 =
3. 20 - 1 = 9 - 7 =
4. 9-5= 10-8 =
5.
10 - 4 =
200 - 1 =
6.
9 - 2 =
9 - 4 = _
7.
30-1 =
10 - 7 =
[3- and 4-place minuends ]
Write the remainders. For help, study the
borrowing in the box.
a b e d
607
- 32
300
-275
5,040
- 706
2,002
-653
800
-89
905
- 640
7,007
- 438
4,000
- 950
350
- 98
200
- 1 09
1,050
- 824
3,003
- 303
500
-409
702
- 300
6,004
-2,306
8,430
- 742
104
- 68
700
- 1 88
5,025
-775
3,206
-3,057
1. Ed’s grandfather showed him this puzzle.
Grandfather said, "Write three numbers of 2
figures each. Then I’ll write three
numbers, and I’ll tell you the
sum of all six numbers without
adding.”
Ed tried to make it hard for
Grandfather. He wrote 79, 87,
and 98.
Quick as a flash, Grandfather
wrote 20, 12, and 1, and said,
"The sum is 297.”
Add in the box. Was Ed’s
grandfather right?
2. Now let’s see how Grandfather chose his
numbers. Look in the next column.
Find the sum of each pair of numbers.
Pairs:
1st
2d
3d
Ed’s numbers:
79
87
98
Grandfather’s:
20
1 2
1
3. Grandfather had
just subtracted each of
Ed’s numbers from
4. Now work in
boxes B and C as
Grandfather would.
If the first three
numbers are 2-place,
the answer is always
297.
79
87
98
710
/2
/
B
24
65
32
C
—
15
Adding and Subtracting Money Numbers
Mary’s bank registers nickels, dimes, and
quarters, up to $10. Every time Mary puts
in a coin and pulls the lever, a bell rings and
the total amount in the bank shows.
1. In the picture, Mary’s bank shows a
total of $ $1.35 is read, “One
and 35 ”
$1.35 could also be written as (ft.
2. If Mary puts in a nickel, the bank will
register a new total of $ If Mary
then puts in a dime, she will have $
in the bank.
3. The addition example for Ex. 2 is shown
in box A. Find the total.
What must you write to show that the an-
swer in box A is a money
number?
4. Mary’s bank will hold
$10. How much more must
Mary add to the total shown
in the picture before the bank
is full?
Subtract in box B.
A
$1.35
0.05
+ 0.10
B
$10.00
-1.35
Find the sums. Check each sum.
a
5. $1.45
+ 1.36
b
$5.08
+ 0.83
c
$0.97
+ 3.06
d
$0.45
+ 0.98
6. $1.50
0.25
+ 0.10
$2.95
0.10
+ 0.05
$0.74
0.05
+ 0.10
$7.50
5.35
+ 1.75
7. $5.95
2.31
+ 1.04
$2.10
0.75
+ 0.25
$0.62
1.18
+ 0.24
$3.50
2.06
+ 1.70
8. $10.25
+ 1.63
$12.78
+ 3.69
$0.38
+ 14.97
$20.10
+ 0.97
Find the remainders. Check.
9. $5.47
-1.24
$1.50
-0.75
$2.08
- 1.70
$6.00
-0.25
10. $16.48
-2.35
$13.60
-5.46
$10.05
-9.02
$8.00
-2.74
11. $4.06
-1.69
$0.84
-0.35
$3.10
-2.08
$9.75
-3.79
12. $ 1 2.0 5
-0.08
$0.98
-0.79
$0.29
-0.04
$40.40
-4.04
Add or subtract as the sign tells you.
13. $ 2 7 . 8 9 14. $ 1 2 . 6 7
+ 24.65 1.02
+ 0.08
15. $3 0.7 0
- 13.80
16. $10.05
0.70
+ 20.86
17. $30.52
-20.95
18. $17.07
- 1 0.49
76
Let's See — How Are We Doing?
A New Kind of Hopscotch
1. The numbers in the curve
at the left end
2. Round each of the numbers inside the
are rounded to the nearest
colored lines in two ways. Use
numbers in the curved ends.
the rounded
In the right end, the numbers
are rounded to
Cross out each number in an
end as you
the nearest
—
use it. 368 is rounded for you.
At the right of each example below, round the numbers and esti-
mate the answer.
Then work the example.
3. Example
Estimate
4. Example
Estimate 5. Example
Estimate
547
500
6,82 1
7,340
+ 288
-4,189
+5,908
Find any mistakes in the examples below. If an answer is right,
make a check mark (v) on the dashed line. Cross out any wrong
answer and write the correct one.
a
b
c
d
e
f
g
h
46
100
$3.06
1,000
$0.68
2 1 7
2,000
$30.17
-39
- 87
-0.85
- 997
+ 0.32
- 1 93
- 897
+ 0.95
1 5
1 3
$2.11
1,003
$0.90
24
1,213
$31.02
Cross out wrong solutions in Ex. 7-9.
7. Mary put a stamp on each of 3 letters.
Then she had 7 stamps left. How many stamps
did she have at first?
(7-3=4) (7 + 3 = 10)
8. Bill had 150 feet of string on his kite.
After he tied on a 50-foot piece of string, how
long was his kite string?
(150 + 50 = 200) (150 - 50 = 100)
9. Ellen needed 6 tablespoons of cocoa for
cookies. There were only 2 tablespoons of cocoa
in the tin.
a. How much more cocoa did she need?
(6+2=8) (6-2=4)
b. She had how much less cocoa than she
needed?
(6-2=4) (6+2=8)
17
Let's See — How Well Do You Remember?
1. If n — 4 = 13, then 4 and
13 are the two parts of n. You
can find n by
and n =
2. If 9 + n = 12, then 9 and
n are the two parts of 12. You
can find n by
from n =
3. 9 + n = 16; n =
4.
n +6 — 12 ;
n =
5.
n - 9 = 9;
n =
6.
7 +n = 14;
n =
7.
n - 2 = 10;
n =
8.
7 +n = 17;
n =
9.
12 = n - 5;
n =
10.
4 +n = 13;
n =
11.
n + 15 = 17
; n =
Add or subtract, as the sign
tells you to do.
a
b
12.
620
78 1
+ 186
- 483
13.
467
2,056
+ 534
-1,877
14.
1,205
15,312
+ 3,078
- 9,423
15.
3,064
32,100
+ 2,078
- 7,905
Solve these problems in your head. Write just the answers.
16. The meeting began at 20 minutes of 2. Mother arrived
at 25 minutes of 2. Was she early or late, and how many
minutes?
17. Ed left at 25 minutes past 3 and rode his bicycle to the
post office in 12 minutes. At what time did he reach the post
office?
18. Ginny had 25^ in her purse after she spent 45^ for a
present for her mother. How much money was in her purse
in the beginning?
i
19. Sam’s hens laid 18 eggs on Monday and 23 eggs on
Tuesday. They laid how many more eggs on Tuesday than
on Monday?
fViwtMeSi
a+td PwffiLeA
There is no trick about this, but it may surprise you!
1. Subtract the numbers in
the box at the right.
2. Now add the figures in
the minuend. Do your work
below.
9 + 8 + 7 + — + : — + — + — + — + — =
3. Now add the figures in the subtrahend.
1+2 + 3 + — + — + — + — + — + — =
4. Add the figures in the remainder.
8 + 6 + 4 + — + — + — + — + — + — =
5. What do you notice about the three sums?
18
A Review of Multiplication
[ Multiplication facts; M. without and with carrying ]
Write the products quickly.
Multiply.
a
b
c
d
e
f
a
b
c
d
1.
2
4
6
9
3
1
13. 6 2
723
822
4,243
X 2
X 5
X 2
xo
>15
X 9
X 4
X 3
X 4
X 2
2.
3
X 8
0
X 1
9
X 5
4
X 8
3
X 3
5
xi
Here is
carrying in
some practice
multiplication.
to help you with
Multiply and add
mentally. For
Ex. 14a,
think , “5 X 2 = 10;
3.
3
2
6
4
7
2
plus 1 is 11.
55
X_7
X 8
xo
X 9
X_5
X 9
a
b
14. 5X2 + 1
/ /
t L
3X9+2
=
4.
2
X 7
5
X8
0
xo
4
X 7
6
X 8
lx
1 00 o
15. 6X4+4
=
7X6+5
=
16. 9X7+5
=
5X9+4
=
5.
9
3
8
7
9
8
17. 8x5+3
=
4X7+3
=
X 8
X 2
X 8
X 9
X 9
X 7
18. 3X7+2
=
7X9+6
=
6.
6
19. 8X6+7
_
9X4+8
_
4
2
9
2
5
X 6
X 6
X 1
X_7
xo
X 6
20. 9x8+7
=
5X3+3
21. 6X9+5
= --
7X7+6
=
7.
6
X_9
7
X 2
0
X 3
4
X 2
8
X 6
1
X 2
22. 7X8+4
= _ _
4X6+3
=
Multiply.
Watch the carrying!
8.
0
7
1
8
3
6
a
b
c
d
X 4
X 1
X 3
X5
X 4
xj5
23. 38 7
763
524
2,56 1
X 6
X 8
X 4
X 7
9.
0
8
7
8
7
6
XJ2
X 4
X 6
xi
X 4
X 3
24. 42 7
5 1 8
825
3,246
X 3
X 5
X 9
X 6
10.
7
4
2
8
1
9
x_o
X 4
^+5
X9
X 6
X 3
25. 269
432
647
1,728
X 4
X 6
X 8
X 8
11.
8
3
7
3
6
5
x_o
X 9
X 6
X 4
X 5
26. 19 8
745
987
4,5 17
X 7
X 4
X 3
X 6
12.
0
6
5
6
9
5
X 5
X_1
X 2
X7
X 2
X 7
79
Multiplying by a Two-Place Number
Multiply. Write the figures of the partial products in neat, straight
columns. Check by doing the work again.
a
b
c
d
e
f
59
67
28
4 1
53
48
X 2 4
X 92
X 2 8
X 6 7
X 35
X 3 9
97
89
76
53
34
98
X 3 6
X 7 5
X 4 8
X 8 6
X 49
X 1 6
684
537
473
365
2 1 9
186
X 3 2
X 2 5
XI 7
X 59
X 92
X 6 4
1. On page 18, you found that 45 is an
interesting number. Here you will find that
37 as a factor gives interesting results.
Do these multiplications:
a. 3 X 37 = , and 1 -4— 1 — (— 1 —
b. 6 X 37 = , and 2 —1— 2 — (— 2 =
c. 9 X 37 = , and 3 —1— 3 — |— 3 =
d. 12 x 37 = , and 4 + 4 + 4=
e. 15 x 37 = , and 5 + 5 + 5 =
2. Give these products without multiplying:
3. Here is another multiplication game.
a. 7 x 15,873 =
b. 14 x 15,873 =
c. 21 x 15,873 =
4. From Ex. 3, write the product for this
example without multiplying:
28 x 15,873 =
5. By what number should you multiply
15,873 to get the product 888,888? Do this
mentally!
a. 18 X 37 =
20
b. 21 x 37 =
x 15,873 = 888,888
Two-Place and Three-Place Multipliers
Miss Otis asked the class to find the product
of 314 X 82.
Bill used the factor 314
as the multiplier.
Ann knew that it is easier
to multiply by the smaller
number, so she reversed
the factors. She also re-
versed the figures in 82,
because she thought that
28 looked like an easier
multiplier than 82.
Sally multiplied by the
smaller factor, 82.
Do all three multiplica-
tions in the boxes.
Put a check (V) beside
the correct answers, and
circle the example that is
in the best form.
Explain what is wrong
with Ann’s way of working
the example.
a
3. 9,84 1
X 4 8
4. 4,3 19
X 6 5
5. 1,782
X 9 1
6. 1,43 5
X 6 8 5
[. Multiplying by the smaller factor ]
b
37
X 8 , 5 9 6
6,785
X 4 6
3,247
X 8 7
4,563
X 3 7 2
Find the products. Reverse the factors when
it will shorten the work.
7. 3,849
X 4 2
576
X 6 , 7 1 8
b
1. 1,24 5
X 15
32
X 2 , 1 3 4
2. 5,786
X 2 5
3,569
X 1 9
8. 4,3 5 9
X 149
8,197
X 2 1 9
21
Multiplying with Money Numbers
1. The baseball outfits worn by players on
the Little League teams in Southville cost
$10.95 each. How much did 18 outfits cost?
You multiply with money
numbers just as with whole
numbers. In the product,
write the decimal point to
show cents and write a dol-
lar sign.
Multiply in the box.
The 18 outfits cost $
$10.95
X 1 8
Find the products. Be sure to write the decimal point and the
dollar sign in the product when you multiply a money number.
a
b
d
2. $ 1 0 . 8 7
X 1 9
$0.75
X 4 0
$25.06 $60.50
X 1 07 X 39
e
$48.34
X 6 8
An estimate of the product is important in multiplying money
numbers. It may show a misplaced decimal point. In Ex. 3 -9,
first estimate; then multiply and write the exact answer.
3.
24 X $16.95 = ?
Estimate:
20 X $20
$
Answer: $_
4.
205 X $9.45 = ?
Estimate:
X
$
Answer: $^
5.
107 X $25.06 = ?
Estimate:
X
$
Answer: $
6.
31 X $83.98 = ?
Estimate:
X
$
Answer: $
7.
952 X $1.16 = ?
Estimate:
X
$
Answer: $
8.
8 X $67.87 = ?
Estimate:
X
$
Answer: $
9.
19 X $0.59 = ?
Estimate:
X
$
Answer: $
10. An estimate is a good check of any product. Notice the zeros
in the rounded numbers: 1,000 X 1,000 = 1,000,000.
A good estimate for the product of 4,625 X 2,132 would be:
X =
22
Practice in Estimating Products
Multiply in your head. Write just the answers.
a b
1. 30 X 600 = 9 X
2. 6 X 800 = 700
3. 200 X 800 = 500
4. 90 X 4,000 = 600
,000 = 100 X 1,000
( 300 = 4 X 900 =
< 500 = 40 X 8,000 =
< 700 = 50 X 900 =
5. To estimate the product of 52 X 49, you can think ,
“50 X = 2,500.”
The exact product is (Do the work in the
box.) Is the estimate close?
In Ex. 6-21, estimate mentally. Write the estimates before
you find the exact products.
Use another sheet of paper for your multiplications.
6. 7 X 69 =
Estimate: 7 X70 =
7. 62 X 51 =
Estimate: 60 X 50 =
8. 9 X 28 =
Estimate: X =
9. 38 X 42 =
Estimate: X - ----- =
10. 21 X 27 =
Estimate: X =
11. 8 X 298 =
Estimate: X =
12. 3 X 587 =
Estimate: X =
13. 60 X 904 =
14. 70 X 508 = _
Estimate: X
15. 58 X 70 = _
Estimate: X
16. 93 X 1,009
Estimate: X
17. 51 X 8,203 =
Estimate: X
18. 82 X 5,600 =
Estimate: X
19. 63 X 3,206 =
Estimate: X
20. 401 X 6,045
Estimate: X
21. 42 X 392 =
Estimate:
X
Estimate:
X
Visiting the Bookmobile
[A., S., M. problems ]
In Stacy’s town, the Bookmobile comes once a week. This is a
truck, full of books on shelves, which the public library from the
city sends each day to a different town.
Write your answers to these problems on the lines provided. Space for Work
Do your work in the space at the right.
1. One morning the Bookmobile loaned 179 books. In
the afternoon it loaned 317 books. That day the Bookmobile
loaned how many more books in the afternoon than in the
morning?
2. In the morning 69 people came, and in the afternoon
127 people. How many came through the day?
3. If 92 people each took 2 books, and 104 people each took
3 books, how many books were taken?
4. Of the 127 people who came in the afternoon, 54 were
children. How many grown people came?
5. The Bookmobile left 28 books at the Hill School, 15 at
the Dale School, and 32 at the South School. How many
books were left at all three schools?
6. After school, Bill went to the Bookmobile. He arrived
at 8 minutes past 3 and stayed for 6 minutes. Then he spent
5 minutes in doing an errand at the store. After that he walked
home, arriving at 27 minutes past 3. How long did it take
Bill to walk home?
7. Stacy returned two books that were 2 weeks late. The
fine was 2^ a day for each book. Stacy had 58^ in her purse.
Was that enough to pay the fine?
24
This Is a Review!
1. In the box you will find an illustration
for each of the terms listed below. Copy in the
blank a number to illustrate each term.
a. minuend
b. factor
c. addend
d. product
e. subtrahend
f. multiplicand
g. remainder
h. multiplier
i. sum
Here are some examples that Miss Otis used
in a test and some of the answers that were
given. If an answer is right, make a check (V)
beside it. If it is wrong, cross it out (X), and
write the correct answer.
2. Round 4,395 to the nearest hundred.
Anne’s answer: 44
3. What is the difference between 200 and
20?
Joe’s answer: 0
4. Give a reasonable estimate for the product
of 23 and 41.
Sam’s answer: 80
5. What is the product of 6 and 9?
Sally’s answer: 54
6. There are 60 seconds in a minute, and
60 minutes in an hour. How many seconds are
there in an hour?
Bob’s answer: 3,600
Draw a circle around A. or S. or M. to show
whether you must add, subtract, or multiply.
Then solve each problem and write the answer.
Sometimes you have more than one step in a
problem and must circle more than one letter.
Work on separate paper if you cannot do
the figuring mentally.
7. Ann spent 20^ and had 30^ left. How
much money did she have at first?
A. S. M. Answer:
8. Bill earns $1.50 each Saturday afternoon.
How much does he earn in 4 weeks?
A. S. M. Answer:
9. Rita had 6 cookies. If she gave one to
each of her three sisters and two to her mother,
how many were left?
A. S. M. Answer:
10. The one-way fare to Kent is 33^, and
the round-trip ticket costs 52 How much
money can be saved on a round trip by buying
a round-trip ticket?
A. S. M. Answer:
11. Ava’s mother put 3-cent stamps on her
Christmas cards. If she sent 86 cards, how
much did the stamps cost?
A. S. M. Answer:
12. The charge for Mr. Rich’s telephone call
was 75^ for the first 3 minutes and 10^ a minute
after that. If he talked for 5 minutes, how
much did Mr. Rich pay?
A. S. M. Answer:
13. Uncle Jack lives 19 miles from his work.
How long is the round trip to and from work?
A. S. M. Answer:
25
1. The number 6,475 has
periods.
a. 6,475 means 6
and 4 and 75
b. 6,475 also means 64
and tens and ones.
c. 6,475 also means 647
and ones.
Round in Ex. 2-5 as directed.
2. 686 to the nearest ten
3. 3,279 to the nearest ten
to the nearest hundred
4. 9,842 to the nearest thousand
5. 653,298,500 to the nearest million
Write in figures:
6. four tens and five
7. six hundreds and six
8. thirty- three tens
9. one hundred thousand, one hundred ten
Write with Arabic numerals:
10. XXIV 12. LXIX
11. XL 13. MCML
14. Write the whole story in A. and S. for
6, 7, and 13.
15. To find n in the example 14 + n =26,
you think , “26 _ _ _ 14 = ”
16. Write two examples in the 5+3 addi-
tion family. ;
17. In the example 3x8, the _ _ means
_ - _ groups, each containing things.
18. If you have to find the product of
2,146 X 35, which number will you use as the
multiplier?
19. Write two examples in the 17—9 sub-
traction family.
20. A good estimate for the product of
8 X 629 is 8 X , or
Find the answers.
a
b
c
d
21.
1 5
+ 34
27
+ 362
$3.26
+ 1.85
5,329
+ 2,972
22.
46
-24
874
- 52
$3.95
-0.66
8,234
-458
23.
206
-39
800
- 356
5,005
- 678
$20.70
- 1 3.85
24. 3 2
86
$3.08
3,926
X 4
X 7
X 6
X 8
25. Write products for Ex. a-c.
Testing What You Have Learned
places and
26
a. 48 X $0.75 =
b. 205 X $4.98 = _
c. 4,507 X 3,060 =
The Flower Shop
Sometimes Iva and Joe help in their father’s
flower shop. One day their father told them to
use roses to show the two kinds of division
(measurement division and fractional-part divi-
sion) which they were studying in school.
He gave Iva 12 roses and some vases, and
told her to put 3 roses in each vase.
The picture below shows that Iva first
counted 3 roses and put them in a vase.
1. Draw another vase beside Iva’s and put
3 roses in it. Keep on drawing vases with 3
roses in each until all 12 roses have been used.
2. How many vases did Iva need for her
roses?
12 -s- 3 =
3. If 12 things are divided into groups with
3 in a group, there are groups.
4. Iva measured 12 by , to see how
many 3’s there are in 12. There are
3’s in 12.
Measurement division tells how many
equal groups are contained in a larger
group.
Joe had 12 roses, too. He was to divide
them equally among 3 vases.
First Joe put 1 rose in each of the 3 vases,
as shown in the picture above.
5. Then Joe put another rose in each vase.
Draw roses in the vases to show this.
6. To show how Joe divided the rest of the
flowers, keep on drawing roses in the vases until
you have used all 12 roses.
7. How many roses did Joe put in each of
the 3 vases?
| of 12 =
8. If 12 things are divided into 3 equal parts,
there are in each part.
9. Joe divided 12 into equal parts,
called One third of 12 is
Fractional-part division tells how
many there are in each of the equal
parts of a group.
10. The division fact 12 4-3 =4 may tell that
a. there are 3’s in 12; or that
b. there are in each of the equal parts of 12.
27
Division Facts and Related Multiplication Facts
1. Complete the following:
a. 2X3= b. 3X2=
< — Quotient — »-
c. 3)6 d. 2)6
2. Ex. 1 tells in different ways that there are
two 3’s in and three 2’s in
3. If 6 = n X 3, then n = To find n
here, you 6 by
We divide to find the missing factor.
Multiplication facts go with related division
facts to make whole stories.
4. The whole story in M. and D. about 2, 3,
and 6 is:
2 X = 6
6 -3 =
3 X =6
6 Pt 2 =
5. Write the whole story in M. and D. about
a. 5, 9, and 45.
b. 8, 9, and 72.
c. 3, 9, and 27. d. 9, 7, and 63.
6. Write each of the following division facts
in two other ways:
a. 9’s in 18 =2
b. 30 : 6 = 5
28
If you know all the division facts, you can
divide any number. Write quotients to finish
the following D. facts, using related M. facts
when they help you.
a
b
C
d
e
7. 5)10
8)64
6)48
9)54
4)28
8. 8)8
6)42"
1)0
7)35
7)63
9. 1)6
4)12"
2)14
3)0
5)20
10. 6)54
7)0
8)16
4)20
9)45
11. 7)7
2)18
5)15
1)T
3)12
12. 8)48
6)36
3)21
6)6
5)25
13. 4)16
4)36
8)24
5)40
4)8
14. 4)0
3)24
5)35
9)18
8)72
15. 7)42
9)9
3)15"
1)8
5)0
16. 6)18
4)24
7)56
7)21
6)24
17. 1)2"
9)63
4)32
6)0
8)56
18. 5)5
9)81
8)40
9)27
7)28
19. 3)18
5)45"
7)49
3)27
8)0"
20. 4)4
7)14
3)9
6)30
9)72
21. 2)16"
9)0
5)30
8)32
1)3
22. 3)3
2)8
1)9
2)0"
9)36
23. 2)6
6)12"
3)6
2)2"
1)4
Table Numbers in Uneven Division
1. Ralph has 110. How many 2^ candies
can he buy? Will he have any money left,
and, if so, how much?
To find out, finish the diagram below.
In uneven division facts, you use table num-
bers to help you.
Under each example, write the table number
you would use. Do not divide yet.
abed
7. 6)25 3)17 4)18 7)32
2. On another paper, subtract to find how
many times you can take 2 out of 11.
There are 2’s in 11, and there
8. 9)52
5)28
8)63 2)15
is a remainder of
B
5, Rl
2jn
C
11 ~2 = 5, Rl
3. In dividing 11 by 2 (boxes A-C), you
have to know how many 2’s you can take from
1 1 all at one time. 11^-2 is not in any divi-
sion table, but 10 -5- 2 is. 10 is the table
number to use because it is next smaller than 1 1 .
10 h-2 = 5, so 11 -2 , R .
9. 3)29 7)46
10. 5)49 9)70
11. 83 - 9 =
13. 25 -T- 7 =
4)31 6)38
2)19 8)47
12. 74 - 8 =
14. 58 -r- 6 = .
4. Complete the table with divisor 2:
2)2 2)4 2)6 2)8 2)10
2)12 2)14 2)16 2)18
5. The table numbers for dividing by 2 are
2 4
, , , j , , .
6. For 2)17 you would use as the table
number because it is next
than 17.
17-5-2 4 R
15. Now divide in Ex. 7-14. Try to think
the multiplication and subtraction and write
only quotient and remainder, as in boxes B
and C. Copy the example on separate paper if
you have to write the work.
16. 26 -4- 4 = , R To check the an-
swer, you multiply X 4 and add
The result should be
17. 65 -9 = I _, R
Check: _ _ _ X9 = ; + _ _ _ =
29
Division with Two-Place Quotient
[Check']
A
B
c
D
34 quotient
26
37, R1
49, R4
Check
^2)68 dividend
2)52
2)75
7)347
49
6 (3 tens X 2)
4
6
28
X 7
”8
12
15
67
343
8 (4X2)
12
14
63
+ 4
divisor
1 remainder
4
347
1. As shown in box D, to check a division
example, you multiply the
and the and add the
, if any. The result
should equal the
2. When you divide by 2, the largest re-
mainder you can have is
3. For the divisor 4, a remainder can be
, , , or _ Ll _ . When the remainder
is , we do not write it.
a
4. 4)1*96
Divide. Study boxes A-D if you need help,
b c d e f
6J2T6 3JTT0 8]60T 7]55T 5)3*23
5. 9)777 3)TT9
7)299* 4)357 8)70*9 6)387
In Ex. 6-8, the answers are wrong. Copy each example in the
space provided and write the work correctly.
2, R6 34, R5 36
6. 2)46 7. 6)239 8. 5)T81
40 21 15
6 29 31
24 30
5
30
1. Write any number — for example, 5,246.
Then, using the same figures, write another
number under the first one, as shown at the
left below. Find the difference between the two
numbers, and show that the difference between
them can be divided by 9 without a remainder.
5,246
- 2,65 4 9)
2. Now you choose a number and work the
puzzle in Ex. 1. Be sure to subtract the smaller
number!
3. Take 11 marbles. Take away 5, add 3,
and the result is 8. Explain.
a
Division with Larger Quotients
Divide. Write all your work,
be d
1. 2)1 ,09 6 8)1 0,62 7 6)1 , 1 9 9 7)4 , 2 5 0
2. 6)7 , 2 5 8 2)1 0,3 5 6 5)6 0 , 1 7 5 3)4 2 , 0 6 1
31
Finding the
See if you can work with averages.
1. Mother said, ‘Til plan an average of
2 sandwiches apiece.” If there were 6 people,
how many sandwiches were needed?
Average
5. Four pupils reported their walking time
between school and the post office. In min-
utes, their times were: Joe, 15; Marge, 18;
Ted, 17; Bill, 14. What was the average time?
2. Bill said, “Maybe we ought to ask each
one just how many he can eat!” This is the
tally (record) Bill made of the number of sand-
wiches each one wanted.
Finish Bill’s table and find the total. Then,
to find the average, divide the total by the
number of addends.
Find the average of each of these sets of
numbers.
Work in the
space below.
a
b
c
d
6. 15
6
2 1 7
4,000
1 9
7
243
3,000
1 8
8
2,000
1 6
Mother . . .
/
Dad ....
//
Sally ....
//
Pete ....
///
Ruth ....
/
Bill
m
Total ....
Average . . .
Was Mother’s plan a good one?
3. Blacken the dashed line below to show the
average length of the four lines under the
number scale.
4. The average of 4, 5, 7, and 8 can be
found by dividing by 4; the average
and
This is a very old Egyptian puzzle:
A mule and a horse were carrying some bales
of cloth.
The mule said to the horse, “If you give me
one of your bales, I shall be carrying as many
as you.”
The horse replied, “But if you give me
one of yours, I shall carry twice as many
as you.”
How many bales was each carrying?
Mule: Horse:
Is there an average in this puzzle?
Why or why not?
is
32
Division by a 2-Place Number
1. When Tim was sick in bed for a long
time, his classmates bought him a bird-
feeding tray for his window. The tray cost
$3.95, and 42 pupils wanted to divide the cost
equally.
Their teacher knew that $3.95 cannot be
divided evenly by 42, so she agreed to pay
the remainder. How much did each pupil pay,
and how much did the teacher pay?
Divide (box A or box B). For 395 42 think ,
“39 tens -f- 4 tens = ”
Multiply and compare. X42 =
Subtract and compare. 395 — =
Each pupil paid _ _ _ ^ ; the teacher,
[ Trial quotient the true quotient ]
a
Divide. Write all your work on this page,
be d
e
2. 22)$ 1.99 30)277 43)777
75)$ 4.5 5 86)703
3. 52)$ 2.25 66)77¥ 90)$ 7.29 64)$ 3.36 45)278
4. 32)278 54)387 21)$ 1.9 6 32)2 3 6 43)77 6
5. 84)778 72)570 56)$ 4.48 63)277 92)783
33
Finding the True Quotient
1. Work the example in box A. First think ,
“19 tens -r- 5 tens.” The
table number is ,
and the quotient figure
is
2. A division example gives you practice in
multiplication, addition, and subtraction.
A
58)T97
a
17. 3 4 6
-328
18. 2 11
- 1 95
[ Trial quotient not true quotient ]
d
b
8 1 2
-810
3 1 8
-276
c
705
-680
262
-252
453
-406
150
-14 1
In the example in box A, you divide 19
by to find the trial quotient; you mul-
tiply 58 by ; in multiplying 3 X 58, you
add to ; then you subtract
from
Here is some practice that will help you
in division examples. Try to do the work in
your head, and write just the
answers.
a
b
3.
5x6+4=
3
X 4 +2 =
4.
6x8+3=
7
X 5 + 3 =
5.
2X8+1 =
4
X
00
+
OO
II
6.
5X9+3 =
9
X 6 + 6 =
7.
4x7+2=
5
X 8 + 2 =
8.
3x9 + 1=
6
X 6 +4 =
9.
7x4+6= _____
9
X
+
00
II
10.
9x5+7= _____
4
X 6 + 1 =
11.
8x3+7=
8
X 5 +6 =
12.
9x4+5=
9
X 2 + 3 =
13.
7X8+5 =
7
II
+
X
14.
6x7+5=
8
X 6 + 2 =
15.
9X8+4=
6
X 9 +2 =
16.
8x7+4=
8
X 2 + 5 =
34
19. For box B, think , “28 tens 3 tens.”
From the division table for the divisor 3, you
see that is the table number, so you try
as the quotient. 9 X 38 = Is
9 the true quotient?
How can you tell?
Try 8. 8 X 38 =
Is 8 right?
Try 7. 7 X 38 =
Work Ex. B. The true
quotient is
Divide. Try to think the multiplication for
each trial quotient.
Trials for Quotient
20. 26JT60 16 -f- 2 =
8 X 26 =
7 X 26 =
6 X 26 =
21. 47}4T6 44 - 4 =
Try 9.
9 X 47 =
Dividing Larger Numbers
' [< Some quotient figures non-apparent]
A
B
C
D
8, R13
21
52, R15
38, R8
36]30T
36)756
45)2355
45)1)718
288
72
2 25
1 35
13
36
105
368
36
90
360
15
8
1. In box A, are there enough hundreds to
give at least 1 to each of 36 groups?
Are there enough tens? Then we
have just 1 quotient figure, and it is writ-
ten in place.
2. In box B, are there enough hundreds to
give at least 1 to each of 36 groups?
Are there enough tens? Then we
have quotient figures. The first one is
written in place.
3. In box C, the first partial dividend is
tens, and the second partial dividend is
ones.
4. In box D, we first try for the ten’s
quotient figure, then For the one’s figure
we try , then
Divide in rows 5-7. Write all your work on this page. Try to
test the trial quotients in your head.
5. 63)2 ,709
45)1 ,2 1 5
73)5 ,110
24)1 ,968
6. 14]38“9
33)1 ,749
65)2 ,360
66)1 ,19 5
7. 79)3 ,250
74)3 ,996
54)3 ,950
58)4 ,060
35
Do you remember Grandfather’s puzzle? (It
is on page 15.) Here is another one.
Grandfather said, "You write any number
of 3 figures, and without dividing I will change
it to a 4-figure number that is exactly divisible
by 9.”
Ed wasn’t sure what "exactly divisible”
means, so Grandfather explained that a num-
ber is exactly divisible by another number
when there is no remainder.
1. Ed wrote 652. Quick as a flash, Grand-
father changed it to 6,525 and said, "There,
6,525 is exactly divisible by 9. Prove it!”
In box A, write the quo-
tient Ed found. Divide on
another paper.
2. Ed wanted to try again, so he wrote 123.
Grandfather changed this to
1,233.
Divide 1,233 by 9. Write
the quotient in box B.
Is 1,233 exactly divisible by 9?
B
9)T72'33
A
9)6 , 5 2 5
3. Grandfather explained Ex. 1 and 2 this
way: "If the sum of the figures in a number
can be divided by 9 without a remainder, the
number is exactly divisible by 9.”
To change 652 (Ex. 1), Grandfather thought,
"6 + 5 + 2 = ” Then from the M.
table for 9’s, he chose the product next larger
than this sum, which is So to make
the sum of the 4 figures exactly divisible by
9, the fourth figure must be 18 — 13, or
4. Why did Grandfather change 123 to 1,233?
5. Change 975 to a 4-
place number exactly di-
visible by 9.
In box C, show that your answer is cor-
rect.
C
91
a
Division with 2-PIace and 3-Place Quotients
[Apparent and non-apparent quotient figures ]
Divide in Ex. a-d below. Write all your work on this page.
be d
1. 34)897
34)8 ,945
47)3 ,307
47) 3 3, 0 7 4
36
6.
2. Notice that Ex. la and lb are very much
alike. In Ex. lb the dividend has one more
figure than the dividend in Ex. la. Is the same
thing true of the quotients?
3. How many figures has the dividend in
Ex. lc? in Ex. Id? How many
figures has the quotient in Ex. lc? in
Ex. Id? ____
291, Rll
13)3394
26
1 09
1 07
24
13
11
4. Miss Otis wrote 42)968 on the board and
said, “How many figures will there be in the
quotient? And tell why.”
Bill said, “Three, because 968 has three
figures.”
Sally said, “Two, because the first division is
96 tens -r- 42. The first quotient figure will be
above 6 in ten’s place, and the second above 8
in one’s place.”
Which was right?
871, R30
7. 35)30,415
28 0
2 41
2 45
65
35
30
Find the mistakes in Ex. 5-8. Then copy
each example in the space at its right and work
it correctly.
120, R51
5. 74)9381
74
1 98
80, R9
8. 24)169
160
9
Divide in row 9. Do all your work on this page,
b c
d
9. 32)6 ,944
53)9 ,5 69
86)8 ,93 7
61)1 4,8 64
How Well Do You Remember?
Can you subtract and multiply correctly? If you can, you will
not have much trouble in working division examples.
Subtract.
a
b
a b c
d
1. 811 300
-759 -202
7,386
-6,427
5,050
-4,863
9. 6,015
X 324
3,456 2,304
X 1 09 X 5 80
2. 70 5 1 40
- 627 - 82
1,005
-868
1,000
-997
3. 4 1 4 749
-236 -708
1,972 3,306
- 1,928 - 3,1 08
10. 8,7 00
X 348
5,170
X 507
8,920
X 570
Multiply.
4. 3 5 4 8 2 6
X 4 X 7
5. 829 3 74
X 5 X 9
473 5,142
X 6 X 8
768 3,475
X 3 X 6
11. As a check to see if an answer is reason-
able, you may estimate the result by using
numbers.
6. 7 1 8 28 7
X 9 X 7
7. 792 1 84
X 54 X 39
496
X 3
5 1 6
X 8 1
2,953
X 6
For each of Ex. 12-16, first estimate the
product. Then write the exact product.
12. 4,625 X 2,132 =
962
X 7 3
Estimated product:
13. 7,800 X $41.50 = _
Estimated product:
14. 1,024 X $31.45 =
8. 5,006 708
X 5 8 X 6 9
3,060
X 37
2,590
X 1 6
Estimated product:
15. 1,932 X 8,239
Estimated product:
16. 86 X $0.69 =
Estimated product:
38
Dividing by a 3-place
divisor is no different from
dividing by any number.
Study the work in the box.
Remember that we must
write a figure in the quo-
tient for each dividend fig-
ure we bring down.
In the problems below, you have to divide by a 3-place Space for Work
number. Use the space at the right to do your work.
1. Green coffee beans are shipped from Brazil in bags con-
taining 133 pounds. About how many bags would 25,000
pounds of coffee beans make?
Why should you round the quotient to the next larger
number of units?
Dividing by 3-Place Divisors
[Apparent and nan-apparent quotient figures]
Dividing hundreds
Dividing tens
Dividing ones
2
20
208, R4
423)88,025
423)88,025
423)88,025
84 6
84 6
84 6
3 42
3 425
3 425
3 384
41
2. A 9,000-acre tract of land was divided into 257 farms.
To the nearest acre, how large was an average farm?
Why would you not round the quotient to the next larger
number of units?
3. Mr. Osgood hoped to average 325 miles a day on a trip
of 1,600 miles. About how many days would the trip take?
Divide in Ex. 4-7. Write all your work on this page.
j 4. 412)8 5,696
5. 224)2 7,300
6. 176)9 3, 1 04
7. 720)2 5,920
Here is another good trick that uses all four
The result in Ex. 1 should be 10. In fact,
operations — addition, subtraction, multiplica-
the result is always 10, whatever number you
tion, and division.
begin with. Now try another.
1. Take any number:
2. Take any number:
Multiply it by 6:
Multiply it by 6:
Add 12:
Add 12:
Divide by 3:
Divide by 3:
Subtract 2:
Subtract 2:
Divide by 2:
Divide by 2:
Subtract the number:
Subtract the number:
Add 9; the result is:
Add 9; the result is:
Using AIS Four Processes
When an example tells you to add, you
know what to do, but in problems you have
to decide what process or processes to use.
In problems 1-7, first draw circles around
the letters that tell the processes you will use.
Then solve the problems mentally and write
the answers. Read the problems carefully!
Some have more than one step.
1. Hedda bought 15 three-cent stamps. How
much did she pay for them?
A. S. M. D. Ans.
2. Amy spent 15^ for 3^ stamps. How
many stamps did she get?
A. S. M. D. Ans.
3. Lucy paid 15^ for a package of enve-
lopes and 3^ for a stamp. How much did she
spend?
A. S. M. D.
40
[ Problems in A., S., M., D.]
4. Joan bought a 15 £ birthday card and a
3^ stamp. The card cost how much more than
the stamp?
A. S. M. D. Ans.
5. Sue wrote 3 notes in 15 minutes. What
was the average time for a note?
A. S. M. D. Ans.
6. One day Mrs. Ames went by bus to
visit a friend who lived on the other side of
town. Each way, she paid 15^ bus fare and
3 for a transfer. How much did the round
trip cost?
A. S. M. D. Ans.
7. Mr. Ames talked for 8 minutes on a tele-
phone call. The charge was 15^ for the first
3 minutes, and 5 for each additional minute.
Find the total cost of the call.
Ans.
A. S. M. D.
Ans.
Units of Measure
1. In the picture you will see many things
that can be measured or that suggest the use
of some unit of measure. Near each of these
pictured items, write the name of the unit of
measure that it suggests.
2. On the dashed lines in the columns below,
write the correct abbreviation for each unit of
measure. Choose from these:
bu.
gal. lb.
oz.
qt.
T.
da.
hr. mi.
pk.
rd.
yd.
ft.
in. min.
pt.
sec.
yr-
feet
rods
gallon _
pint _ _ _
pecks
miles
day
ounce
minute
ton
inches
pounds
yard
quart
second _
hour _ _
bushel _ _
foot _ _ _
17. Write three units
of meas
following kinds of measurement.
[ Tables ; abbreviations]
From the list of units of measure in Ex. 2,
choose and write the one that fits best in each
of Ex. 3-16.
3. Mrs. Day’s family uses 14
of milk a week.
4. Betty’s weight was 75
5. The pencil was 5 long.
6. Bill lives 2 from school.
7. Sarah’s letter weighed 2
8. The baseball game was 2 long.
9. Mrs. Leeds bought 4 of meat.
10. Each member of the spelling team was
allowed 15 for each word.
11. Tom’s father ordered 3 of coal.
12. The tank in Mr. Jones’s automobile holds
15 of gasoline.
13. Jack’s pulse (heartbeat) was 78 beats per
14. Sam’s father is 72 tall.
15. The telephone pole was 25 high.
16. The roadside market sold apples at 3
for 29
Measures of
Measures of
Linear Measure
Liquid
Dry
Time
Weight
(Distance or Length)
Measure
Measure
41
Changing from One Unit of Measure to Another
1. There were 48 candy bars in a carton.
Each bar weighed 1 oz. How many pounds
of candy were there in the carton?
GGGGGGGGGGG&
GGGGGGGGGGGG
GGGGGGGGGGG
GGGGGGGGGGGG
Draw a box around enough candy bars to
make a pound. Keep on drawing boxes until
you have used all the candy bars.
48 oz. = lb.
2. To change 48 oz. to pounds, you must
48 by Since pounds
( multiply ; divide )
are than ounces, the number
( larger ; smaller)
of pounds in 48 oz. is than 48.
{more; less)
3. Mr. Storrs had 2 bu. of grain. He had
how many pecks?
On the lines, draw pictures of the pecks in
each bushel.
2 bu. = pk.
4. To find how many pecks there are in
2 bu., you _____ 4 by
{multiply; divide)
Since pecks are __. than bushels,
{larger; smaller)
the number of pecks in 2 bu. is
{more; less)
than 2.
5. The number line below represents 48 inches. Mark it to show
that 48 in. = ft.
i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i r- i i i i i i i i i i i i i i i i i
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
6. The number line below represents 3 yards. Mark it to show that
3 yd. = ft.
r
0
T
2
3
Here is some practice in changing from one unit of measure to another.
To change Remember that
7. 3 feet to inches 1 ft. = in.
8. 14 days to weeks 1 wk. = da.
9. 6 yards to feet 1 yd. = ft.
10. 4 pounds to ounces 1 lb. = oz.
11. 28 pecks to bushels 1 bu. = __ pk.
12. 180 seconds to minutes 1 min. = sec.
42
Multiply
or divide?
Result
12 by
3 ft. = iib in.
. 14 by
14 da. = _ wk.
_ 3 by
6 yd. = ft.
. 16 by
41b. = oz.
28 by-...
28 pk. = _ - bu.
- 180 by__-
__ 180 sec. = - m
Adding and Subtracting Measurement Numbers
2 ft. 8 in.
^-11
3 ft. 4 in.
2 ft. 3 in.
1. In box A, find the total length from the back of the trailer
to the end of the cart’s handle.
The sum in the inches’ column is inches. So you
change inches to feet inches, and carry
feet to the column for feet. Finish the work.
2. In box B, find how much shorter the body of the trailer
is than the body of the cart.
You cannot subtract 8 from 4, so you borrow foot
from feet. This gives you inches to add to
inches, and you subtract 8 from Finish the work.
3. In the examples on this page you must know these measures:
inches (in.) = 1 foot (ft.)
feet (ft.) = 1 yard (yd.)
^ ounces (oz.) = 1 pound (lb.)
_ _ _ months (mo.) = 1 year (yr.)
Add. Be careful about the carrying.
4. 6 ft. 9 in.
+ 4 ft. 6 in.
5. 1 0 lb. 3 oz.
+ 9 lb. 14 oz.
6. 2 yd. 2 ft.
+ 3 yd. 1 ft.
seconds (sec.) = 1 minute (min.)
minutes (min.) = 1 hour (hr.)
quarts (qt.) = 1 gallon (gal.)
pecks (pk.) = 1 bushel (bu.)
7. 5 hr. 3 5 min. 1 5 sec.
+ 2 hr. 10 min. 5 0 sec.
8. 1 1 yr. 7 mo.
-f 1 0 yr. 8 mo.
9. 7 bu. 3 pk.
+ 5 bu. 2 pk.
10. 2 gal. 2 qt.
+ 3 gal. 1 qt.
11. 1 yd. 2 ft. 9 in.
4- 2 yd. 1 ft. 8 in.
Subtract. Remember that in borrowing you change one larger unit to smaller units.
12. 8 min. 1 0 sec.
— 5 min. 4 0 sec.
13. 9 ft.
- 4 ft. 8 in.
14. 1 6 lb. 4 oz.
- 1 0 lb. 12 oz.
15. 4 yd. 2 ft. 6 in.
- 3 yd. 2 ft. 1 0 in.
16. 7 yd. 1 ft.
— 4 yd. 2 ft.
17. 5 gal. 2 qt.
— 3 gal. 3 qt.
18. 3 bu. 1 pk.
— 1 bu. 3 pk.
19. 3 hr. 5 min. 5 sec.
— 1 hr. 2 0 min. 3 0 sec.
43
Multiplying and Dividing Measurement Numbers
The girls in the Home Crafts Club made a
knitted afghan to show at the County Fair.
Here are some of the problems the girls had
when they were making the afghan. For each
of problems 1-4, write the letter of the box that
shows the work and copy the answer.
1. Mary knitted 4 short strips, each 1 ft. 4 in.
long. What was the total length of her knit-
ting?
Box: Answer:
A
Side work
1 ft. 4 in.
X 4
4 x 4 in. =
in.
5 ft. 4 in.
16 in. = 1 ft.
in.
B
5 in.
9)3 ft. 9 in.
= 9)45 in.
45 in.
C
1ft.
4 in.
4)5 ft. 4 in.
4 ft.
1 ft. 4 in.
1 16 in.
16 in.
D
9
3 ft. 9 in. -f-
5 in. = 45 in
. -4- 5 in.
5)45
45
2. If each strip is 5 in. wide, how many
strips, sewn together, would make 3 ft. 9 in.?
6. Show below how we get 45 in. in box B.
Box: Answer:
3. To make a strip running the full length
of the afghan, 5 ft. 4 in., the girls sewed 4
equal short strips together. Find the length
of each of the short strips.
Box: Answer:
7. In box B, if the divisor were 3, would
you still have to change 3 ft. 9 in. to inches
before dividing?
8. In box D, if the divisor were 3 in., would
you still have to change 3 ft. 9 in. to inches
before dividing? Explain.
4. What is ^ of 3 ft. 9 in.?
Box: Answer:
5. Finish the side work in box A. The an-
swer in box A is ft. 4 in. and not 4 ft. 4 in.
9. In box D, does the quotient, 9, mean
“feet” or “inches” or “strips”?
Why?
Why?
44
Multiplying and Dividing Measurement Numbers
Write only your answers here. Try to do Ex. 1-6 in your head.
a
1. 1 yd. 2 ft.
X 3
b
2 ft. 5 in.
X 5
c
3 hr. 15 min.
X 4
d
2 qt. 1 pt.
X 8
2. 3 pk. 4 qt.
X 3
4 lb. 7 oz.
X 5
8 gal. 2 qt.
X 2
1 ft. 6 in.
X 6
3.10 min. 2 0 sec.
X 2
6 bu. 2 pk.
X 7
2 hr. 4 5 min.
X 2
2 yd. 2 ft.
X 5
4. 5 min. 3 sec.
X 4
5 lb. 3 oz.
X 4
5 gal. 3 qt.
X 4
5 pk. 3 qt.
X 4
5. 3)1 2 gal. 3 qt. 5)3 ft. 4 in. 3)1 hr. 4)6 lb. 4 oz.
6. 4)2 yd. 2 ft.
6)7 bu. 2 pk.
4)1 0 min. 2 0 sec. 3)4 wk. 2 da.
b
c
7. 15 minjr hr. 1 5 min.
2 da.)3 wk. 1 da.
4 oz.)2 lb. 4 oz.
8. 1 pt.)2 qt. 1 pt.
2 ft.32~-ycL
3 qt.)5 gal. 1 qt.
9. 2 in.)3 ft. 4 in.
5 sec.)5 min. 5 sec.
2 ft.)3 yd. 1 ft.
10. To divide 2 yd. of ribbon into 6 equal
parts, you must change 2 yd. to ft. and
find of that number.
^ of 2 yd. = ft.
11. To find how many 15-minute radio pro-
grams you can hear in 1 hr. 45 min., you change
1 hr. 45 min. to min., and divide
by 15. The answer is programs.
45
Perimeter and Area
[Square inch; square foot ]
1. Which figures are squares?
2. List the other rectangles.
3. How many small squares in each figure?
A B C D
E F G
4. Figures with the same area are and
; _ _ _ _ and ; and
5. A square and another rectangle have the
same area if they contain the same number
of units of the same
kind.
6. Two figures may have the same
even if they do not have the
same shape.
Draw the following figures below. Label the width and length of
each. Let each small square mean 1 square inch.
Below each figure, write its area (A) and its perimeter (p).
7. A square 5" on a side.
8. A rectangle 4" by 6".
9. A rectangle 2" wide that contains 8 square
inches.
10. A square that contains 9 square inches.
11. A rectangle 4" long with a perimeter of
14". (Helper. The two 4-inch sides will use
8 inches of the perimeter.)
11.
10.
9.
8.
7.
A =
sq. in.
A =
_ _ sq. in.
A =
sq. in.
A =
sq. in.
A =___
_ sq. in.
P =
in.
P = --
in.
P =
in.
P =
in.
P = -
in.
c. The area of the afghan was sq. in.
d. Since sq. in. = 1 sq. ft., the
area of the afghan in square feet was
sq. ft.
46
12. The afghan made by the Home Crafts
Club (page 44) was 3 ft. 9 in. wide and 5 ft. 4 in.
long.
a. It was in. wide.
b. It was in. long.
Areas and Rectangles
The rectangle above represents the afghan
made by the girls in the Home Crafts Club (see
page 44). You are going to complete the dia-
gram so as to show the pattern of the finished
afghan.
Each small division of the squared paper
stands for 1 inch, so each small square stands
for 1 square inch.
1 . The afghan was 5 ft. 4 in. long and 3 ft. 9 in.
wide. Finish the marking that is started at
the top and the right side of the afghan, to
show the dimensions in feet.
2. To make the afghan, the girls knitted
strips 5 inches wide. In the diagram, draw
horizontal lines (lines running from left to
right) to show these 5-inch strips. Use your
ruler to help you keep the lines straight.
3. You should have strips, each
squares wide.
4. At the left of the rectangle, label the
strips. Beginning at the top, letter the first
strip A, the next one B, and so on. The strip
at the bottom should be lettered I.
5. Make small marks on the lower edge of
the diagram to show how each strip was divided
into blocks V 4" long.
6. Draw vertical lines (from top to bottom)
to divide the whole rectangle into blocks 1 ' 4"
long.
7. In strip A, beginning at the left, color
the first and third blocks blue.
8. In strip B, color the second and fourth
blocks blue.
9. Continue to color the blocks so as to
make a checkerboard pattern. Strips C, E, G,
and I should be like A; and strips D, F, and
H like B. When you have finished, your dia-
gram will show all the parts of the afghan in
their correct relationship to one another.
10. Each block is 5 in. wide and in.
long. What is the area in square inches of
one block? - of all 36 blocks?
Does this agree with your
answer to Ex. 12c, page 46?
47
Measuring Time by the Clock
[Second; minute; hour; day ]
1. Clocks and watches measure time in sec-
onds, , and
2. Complete this table of time:
seconds (sec.) = 1 minute (min.)
minutes (min.) = 1 hour (hr.)
hours (hr.) = 1 day (da.)
3. In clock A, the minute hand points to 5;
why does it mean 25 min. past the hour?
4. “Half past 3” means 30 min. past 3.
Why?
5. What is another way of saying that the
time is quarter past 2?
6. You could read the time on clock A
as min. before (or “of”) , but
usually it is better to say the number of minutes
before or after the nearer hour.
7. Below each of clocks A-E write the time
the clock tells.
8. Travel timetables use a short way of
writing clock time. “Twenty minutes past
5” may be written 5:20, which is read “five
twenty.”
So 4:45 means min.
4 o’clock, which is the same as minutes
before o’clock.
9. Most clocks measure 12 hours, so the
hour hand must go completely around the
clockface times in 1 day.
48
10. a.m. means time from midnight to noon;
p.m. means time from noon to midnight. Look
up these abbreviations in your dictionary and
write in your own words what the initials
stand for.
11. Draw hands on clocks F-J, above, to
show the time given under each clock.
Measuring Time by the Calendar
[Week, month, year, century ]
C Time by the Calendar
days (da.) = 1 week (wk.)
wk. = 1 year (yr.)
months (mo.) = 1 yr.
1 mo. (except February) = or days
February has da. in a common year, and
da. in a leap year
yr. = 1 century
1. Finish tables A, B, and C.
D NOVEMBER 1Q
Sun.
Mon.
Tues.
Wed.
Thurs .
Fri.
Sat.
5. Look at a calendar for December, this
year. Did you answer Ex. 4 correctly?
2. Look at a calendar to find on what day
November begins this year. Then finish the
calendar (box D) for November.
3. On your calendar, make circles around the
dates for Veterans’ Day (Nov. 11) and Thanks-
i giving Day (which is the fourth Thursday in
November).
I 4. Use your November calendar (box D) to
figure out the day of the week on which each
of these dates will come:
I
j a. Dec. 25, this year.
b. Jan. 1, next year.
1
I
6. Sometimes we write dates in a short way.
For example, March 14, 1956, may be written
3/14/56, or 3-14-56.
Write these dates in the long way:
a. 4/1/55
b. 6-20-56
c. 12/25/57
d. 1-3-42
7. Write the month, day, and year of your
birth in the short way of Ex. 6.
49
a
1. $3.24
1.08
+ 0.62
2. 480
- 1 75
3. $9.08
-8.09
a
4. 8 7
X 4 5
a
5. 25)$ 6.75
6. 300)2 ,1 0 0
How Well Do You Remember?
Work the following. In division,
show remainders with R.
[ Review ]
b
c
d
e
f
nr
&
53
1 89
+ 796
3,097
45
+ 492
485
5,2 10
+ 86
756
207
+ 1,464
$50.75
8.09
+ 23.45
21,986
6,009
+ 783
875
- 698
$10.98
-8.89
6,463
- 908
$4.50
-2.75
7,100
-4,827
35,296
- 26,5 49
736
+ 684
5,347
+ 965
7,2 13
-472
$11.06
-3.28
$80.25
+ 29.87
76,124
- 33,29 1
b
c
d
e
f
54
X 2 0
208
X 3 7
$3.52
X 2 4
6,248
X 5 3
4,302
X 6 3 4
b
c
d
72]4TT
30)$ 2 5 0.50
24)$ 12,296
236)3 1,094
518)2 3 6,406
482)9 8,372
50
Testing What You Have Learned
Add or subtract.
a
b c
d
e
f
1. 3 76 $2.95 $25.00
+ 29 5 + 1.98 - 9.75
$9,500 32,986
- 4,67 5 + 1 5,2 1 5
54,209
-18,064
Multiply.
2. 273
X 1 5
b
$8.69
X 3 2
c
2,195
X 5 4
d
18,962
X 2 6
Divide. Write any remainder, with R, beside the quotient.
3. 42)2 ,5 5 6
57)1 ,254
77)4 ,928
65)5 ,885
Add or subtract. Be careful when you carry and borrow.
4. 3 ft. 8 in.
+ 2 ft. 9 in.
6 hr. 35 min.
+ 7 hr. 45 min.
4 lb. 10 oz.
+ 5 lb. 12 oz.
3 qt. 1 pt.
+ 2 qt. 1 pt.
5. 8 min. 50 sec.
— 5 min. 38 sec.
2 bu. 2 pk.
1 bu. 3 pk.
10 yd. 1 ft.
- 7 yd. 2 ft.
8 ft. 9 in.
- 7 ft. 10 in.
Multiply. Try to carry mentally.
6. 2 ft. 7 in.
X 4
4 hr. 20 min.
X 5
2 yr. 8 mo.
X 3
3 qt. 1 pt.
X 6
Divide. Write just your answers on this page.
7. 4)9 yd. 1 ft.
12)9"Tir:
8 in.)6 ft.
3)7 lb. 2 oz.
8. Round 875,296 a. to the nearest ten ; b. to the nearest thousand
9. Write in Arabic numerals: a. MDCIV ; b. XCII
[Cumulative Review ]
g
57,350
- 29,48 7
e
$302.79
X 1 2
38)7 ,994
6 gal. 3 qt.
+ 3 gal. 2 qt.
4 wk. 3 da.
— 2 wk. 5 da.
8 lb. 13 oz.
X 2
3 pt.)15 gal.
57
Reviewing Fractions in
Good cooks use a lot of fractions! Let’s see
what you remember about fractions.
This is a table of standard cooking measures.
All measurements are level.
3 teaspoons (tsp.) = 1 tablespoon (tbsp.)
16 tablespoons = 1 cup (c.)
2 cups = 1 pint (pt.)
2 pints = 1 quart (qt.)
Cooking Measures
To measure accurately, most cooks use measuring
cups and spoons like the ones pictured above.
Draw pictures below to show how many of the smaller measures
are needed to equal one of the larger. Fill in the dashed lines of the
first small measure and draw as many more small measures as are needed.
The table of measures above will help you.
1. 1 pint = cups
(Ex. 1)
2. 1 cup = tablespoons
(Ex. 2)
3. 1 tablespoon = teaspoons
(Ex. 3)
4. Mark the teaspoons below to show:
5. Since there are tsp. in 1 tbsp.,
a. J’s. Color or shade | tsp.
b. i’s. Color or shade \ tsp.
c. J-’s. Color or shade | tsp.
a.
(o
b.
d
c.
d
tsp. = § tbsp.
6. 1 cup =16 tbsp., or tsp.
7. \ cup = tbsp.; \ cup = tbsp.;
J cup = tbsp.
Most measuring cups are marked to show both thirds and fourths
of a cup. Finish Ex. 8-1 1 below by expressing each amount as a fraction
of a cup. Then finish the diagram for each exercise by coloring or
shading the correct part of the cup.
(Ex. 8) (Ex. 9) (Ex. 10) (Ex. 11)
10. 8 tbsp. = _ _ _ cup
8. 4 tbsp. = __ cup
52
9. 2 tbsp. = cup
11. 16 tsp. = ___ cup
Meaning
1. In this square, of the equal
parts are colored. The fraction that tells what
part of the square is colored is
2. If a thing is divided into [___
fourths, there must be parts, and all the
parts must be
a Fraction
[Terms; writing fractions']
3. In the fraction f, the numerator is ,
and the denominator is
4. A fraction is a number that means one
or more of a whole.
5. Under each of the diagrams below, write
the fraction that tells what part of the diagram
is colored.
6. Beside each of these fractions, write a
letter to tell which diagram at the right shows
that fraction as colored. Be careful! Some
diagrams do not fit.
3 2 3 5_
8 3 5 6
5 3 3 1
9 4 10 2
7. Color or shade one of the pictures at the
right to show each fraction listed below. Then
beside each fraction below write the letter of its
diagram.
3 5
10 8
2 1.
9 4
! 4
6 7
4 1
5 3
Write in figures:
8. Three eighths __
9. Two thirds
.
10. One fourth
11. Four fifths
12. Nine tenths
13. One half
14. Five sixths
15. Two sevenths _ _
Write in words:
16. *
17. |
18. f
19. *
53
Fractional Unit
1. Circle A is divided into equal parts.
The size of one of these parts is the size of the
fractional unit for circle A. The fraction that
names this fractional unit is
2. The fractional unit for circle B is
Color or shade J of circle B.
3. Draw one more line in circle C to finish
dividing the circle into thirds. On each of the
equal parts, write its name.
4. Divide circle D into halves. There are
equal parts. The fraction that names
the fractional unit is
5. In the fraction J, the numerator is ;
the denominator is
6. Circles A, B, C, and D are equal in size.
Circle has the smallest fractional unit.
Circle has the largest fractional unit.
7. From Ex. 6, you see that the denominator
gets _ . as the size
{larger; smaller)
of the fractional unit gets
{larger; smaller )
Another way to say this is: The fewer equal
parts into which you divide a whole, the
each part will be.
( larger ; smaller)
8. a. In circle A there are ^’s.
b. In circle B there are J’s.
c. In circle C there are three
d. In circle D there are |-’s.
[Meaning; size ]
a.
b
9. Look at squares E, F, G, and H, above.
Under each square, write
a. the name of the fractional unit that meas-
ures the square;
b. the part of the square that is colored.
10. The of a fraction
shows the size of each equal part, that is, the
size of the fractional unit; the
tells the number of equal parts in the
fraction.
11. In the fraction f each equal part is
of the whole, and there are of the
equal parts. The fractional unit for § is
12. Under each fraction below, write
a. the fractional unit;
b. the number of equal parts in the fraction.
3 7. 4 _9_ 5. 1_
4 8 5 10 9 6
a.
b.
13. The fractions below are all parts of the
same whole. Circle the larger fraction in each
pair.
a.
1
2
1
100
d.
3
4
3
8
b.
1
7
1
9
e.
5
16
1 1
16
c.
2
7
4
7
f.
5
1 6
5
11
54
Fractions on a
1. The fraction § is a proper fraction. In
a proper fraction the
is less than the
2. The fraction § is an improper fraction
because the is
greater than the
3. The mixed number 2\ means
plus A mixed number is the sum of a
number and a
4. Copy these numbers in the columns be-
low to show what kind of number each is:
6
5
3
334
217
1 6
A7
1
2
80
7
4
4
5
1 9
16
->4
23
8
6A
1,346
75
100
Whole
Mixed
Proper
Improper
Numbers
Numbers
Fractions
Fractions
Number Line
[Proper and improper fractions; mixed numbers ]
A B
0 \ 1 l.| 2 ' 2g 3 3g
On the number line above, each number
tells the distance on the line from 0 to that
number.
5. A marks the distance from 0 to ;
B marks the distance from 0 to .
6. With arrows and letters, show these dis-
tances on the number line above:
C, 2\ D, If E, 3J F, i
i
i
i
1 1 1 j
0
_r i 1 ] 1 i 1
i
p i 1 1 1 1 1
2
The diagram above shows part of a ruler.
Each inch is divided into J-’s, s, and J-’s.
7. On line X, show a distance of 1^ in.
8. On line Y, show a distance of 2f in.
9. On line Z, show a distance of § in.
Equal Parts of a Group
A0®0 BD □ □
° □
□ □ □
o
°oo°
c A
AA
A A
AAAAA
1. In A, there are small circles,
and are colored; so f of the group of
circles is colored.
2. In B, of the group of squares
is colored.
3. In C, there are triangles in all,
and triangles are colored. Each triangle
is what part of the group? What part of
all the triangles is colored?
4. The fraction § may mean of the
equal parts of one thing, or it may mean
of the equal things (parts) in a
group of things.
55
Familiar Fractions
1. Bar X stands for 1, or a whole thing.
The other bars are divided into different num-
bers of equal parts. Above each bar, write the
name of the fractional unit.
a. There are J’s in 1.
b. There are ^’s in 1.
c. — — = 1 d. —7“ =1 e. -77“ = 1
4 6 16
f. — = i g.^ = i
2. Circle the larger fraction in each pair.
a.
i_
2
i_
4
d.
1
4
1
8
[Comparing fractions ]
3. Write the missing numerators. If you need
help, use the bars at the left.
II
Tf | 00
cd
2
b.| =
8
i
c-i = "
A 2
d* 6 =
~3~
3
e‘8 =
l6~
f .12 J
16
4. Copy the fractions in each group in order
of size, beginning with the smallest.
a.
3?
111
123 83 4
K i i
83 23 163
1.
6
r _1_ _1_ 1 1
123 163 43 2
5. Label bars Y and G below to show the size
of each of the equal parts.
6. Supply the missing
numbers below.
2
a* 10 “ 5
1 2
7. Draw a circle
equal to
around
each
of the fractions below
that
are
3
4
12 1 3
16 3 8
7
1 2
8
16
5 2 5
8 4 12
3
6
2
3
6 4 71 4 3 6 81 7 2 _7_ _3_
16 6 8 4 8 16 8 12 8 16 6 10 10
8. Draw a line under each of the fractions above that are greater
than
9. Draw a square around each of the fractions above that are less
than
56
A Fraction Means Division
1. Column (1) lists different
ways of reading some fractions.
Write the fractions in the usual
way in column (2).
(1) (2)
a. three J’s
b. 7 tenths
c. 4 -r- 3
d. ten sevenths
e. 7)10
f. three fourths
g- 4J3
h. 4 thirds
i. 7 + 10
j. ten y’s
k. 10)7
l. four J’s
m. 3)4
n. 3-f-4
o. seven ^’s
p. 10 - 7
3. Suppose 5 stamps are divided equally between 2 girls.
a. Each girl can have stamps, and there will be
stamp left.
b. Box shows the division.
c. Why isn’t it sensible to give the answer as 2\ stamps?
2. Suppose 5 cookies are divided equally between 2 boys.
a. In box C, draw the cookies that ^
each boy can have.
b. Each boy can have cookies.
c. Box shows the division.
4. Write a problem about 9 : 5 in which you would give
the answer with the remainder in a fraction.
Divide, and show any remainder in a fraction.
5. 7j25 6. 4)3" 3 7. 5)42
8. 8)2~24
9. 6)48T 10. 23)88'
11. 3)26" 12. 9)5”8 13. 4)3“9
14. 9)6~0“8
15. 7)6T5" 16. 50)4 ,600
57
The Golden Rule of Fractions
1. If you cut this pie into 6 equal pieces,
the fractional unit is
2. If the pie is cut into 3 equal |v^
pieces, the fractional unit is
3. J of the pie is than J.
{more, less )
4. | of the pie is the same as
third of it.
5. Of the equal fractions § and J,
a. has the larger fractional unit.
b. has the smaller denominator.
6. The larger the fractional unit is, the
the denominator is.
{smaller, larger )
7. In the fraction f, the terms are
and
8. You can change § to ^ without looking
at a diagram if you divide both the 2 and the
6 by
[ Changing to higher or to lower terms ]
12. Balance each pair of scales in Ex. b-h
below by writing in the empty circle a number
equal to the number in the other circle. Choose
the numbers from the ones given in the colored
circles. (The scales in Ex. a are balanced for
you.)
Best Form for a Fraction
To be in best form, fractions in an-
swers should be in lowest terms.
A fraction is in lowest terms when no
number except 1 will exactly divide both
its terms.
9. You can change J to the equal fraction
f if you both terms by
The Golden Rule of Fractions
Multiplying or dividing both terms
of a fraction by the same number
does not change the value of the frac-
tion.
10. The fraction ^ is in
{lower, higher )
terms than the equal fraction f .
11. Using the Golden Rule of Fractions, write
the missing numerators and denominators in the
fractions below.
2 6 8
a.
5
25
10
100
b.
5
6 -
10
II
in j
II
CN
II
36 ~ 30
58
Reduce the fractions below to best form.
Write the number by which you divide both
terms. Think the divisions.
a
b
C
13.
Original fraction:
3
12
6
16
8
28
Divide both terms by:
Fraction in best form:
a
b
C
14. |
Original fraction:
4
12
12
16
18
32
Divide both terms by:
Fraction in best form:
— J
Changing Improper Fractions
[To best form]
The Boys’ Club used 30 rolls at their picnic.
How many dozen rolls were used?
1. How many are there in 1 doz.?
2. One roll is of 1 doz. rolls.
3. 30 rolls are y^’s of a dozen, or ff
of a dozen.
4. In the diagram, draw a ring around each
yf to find how many dozen in ff .
a. There are 2^- rolls.
b. 6 rolls are dozen, so
2— dozen = 2— dozen.
5. The improper fraction yf is in best form
when it is changed to the mixed number
6. To change an improper fraction to a
whole number or a mixed number, divide its
by its
Numbers are in best form if improper
fractions have been changed to whole
numbers or to mixed numbers, and if
proper fractions are in lowest terms.
Change to best form. Show all the steps, but divide on separate paper.
7.
18
8
12.
II
81^
17. f =
8.
II
13.
II
i8. ~ =
9.
II
14.
->12
II
19. f =
10.
100
100 ~
15.
II
20- f-
11.
329
4 “
16.
126
8 “
21.
0
59
Adding and Subtracting Like- Fractions
A
B
3
8
9
10
+ i
3
10
10 _ 1 2 _ 11
8 ” 1 8 ~
6
10
_ 3
— 5
1. Box A. The fraction f means
i’s, or of the 8 equal parts of a whole.
2. In f, the fractional unit is
3. f and are called like-fractions because
they have the same fractional unit, as shown by
their
4. 3 eighths + 7 eighths = eighths.
10 eighths written as a fraction is ,
which in best form is
Fractions with the same fractional
unit are called like-fractions.
5. Box B. and fy are like-fractions because
the fractional unit for both is
6. 9 tenths — 3 tenths = tenths, which,
as a fraction in lowest terms, is
We can add or subtract like-fractions,
that is, fractions with the same fractional
unit.
7. From boxes A and B, copy
a. three proper fractions
b. an improper fraction
c. a mixed number
Read each problem carefully. Then circle “A.” or “S.” to show
whether you will add or subtract. Write your work in the space at
the right of the problem, and then write your answer on the line.
8. One candy bar weighed f oz. and another weighed § oz.
How many ounces did the two together weigh?
A. S. Answer:
9. Joan used f yd. of cloth for the skirt of an apron, and
| yd. for pocket and ties. How much cloth did she use?
A. S. Answer:
10. In Ex. 9, how much more cloth did Joan use for the skirt
of the apron than for the pocket and ties?
A. S. Answer:
11. Jack washed storm windows for fhr., spent Jhr. in
pruning the rosebush, and raked leaves for § hr. How long
did he work in all?
A. S.
60
Answer:
Adding and Subtracting Mixed Numbers
[ Like-fr actions without and with carrying and borrowing]
A
B
c
D
E
F
2ft
3f
4f
61
8 =
7|
n = 8i
+ 4 A
+ 5|
+ 2f
— 4|
-3| =
3|
- 4| = 4|
6ft =
6f
8f
= 9f
6¥ = 7f = 7|
21
= 21
4i
4f
= 41
The examples in the boxes will remind you
Watch the carrying in addition and the bor-
how to add and subtract mixed numbers.
rowing in subtraction.
Add. Always change
answers
to best form.
a
b
c
d
e
1. 4 §
2 f
5ft
1 i
1 2
6 t
+ 3
+ 4|
+ 1 A
+ 2 \
+ 2 f
2* 3 ^2
8
3 i
2 ^
Z 9
2ft
+ 2 A
+ ift
+ 3J
+ 41
+ 3 ft
3. 9*
i
6
2 t
7 i-
• 6
i A
+ f
+ l|
+ 6 f
+ 2
+ 7H
Subtract. Be sure your answers
are in best form.
4. 7 ^
8I
9
8 |
11-1
8
4
6
4
-4|
— 3 |
- 2 -
-4?
8
4
5
6
4
5. 6
4-1
2 ~
5 i
4
8
16
6
-4!
— 2 5
- 2 H
- 2 ^
- 3 -
3
8
16
6
7
6. 9 1
eg
7 —
4 -+
5 —
8
32
12
16
10
— 4 |
— 3 —
- 2 —
— 3 —
8
32
12
16
10
61
Adding and Subtracting Unlike- Fractions
[ Common denominator present; carrying and borrowing']
A
2 _ 4
3 — 6
l 5 5
i 6 6
B
4| =4f
+ lf = If
c
31. — OX
~ Z6
U li _ 13
i2 — A6
D
8 = 7A-6
0 '16
_ Z-5 Z 5
-T6 ~ -'ig
9 _ 13 _ 11
6 1 6 -*• 2
5¥ =6|
14 _ 12
A 6 ~ a3
oli i
^16
0 12 3
1. Show the addition of § and f on the
number line above, as follows:
a. Draw an arrow, lettered A, to show the
point § to the right of 0.
b. Draw an arrow, lettered B, to show the
point that is f to the right of A.
c. Your arrow B points to 1— — > or 1-y-
2
2. The point A shows that - = -y-
4. The common denominator of f and f
is The common fractional unit is
Fractions with a common fractional
unit have a common denominator.
In each pair change one fraction so that the
fractions will have a common denominator.
5.
l
4
5
8
7.
3
5
2
10
8.
5
6
1
3
1
8
10.
2
16
3
4
3. | and f are -fractions;
| and f are -fractions.
11. In Ex. 5-10, did you change to the larger
or the smaller fractional unit?
Add or subtract. First be sure that you understand the work in boxes A-D above.
12.
13.
14.
7
8
2
16
2
3
1
6
15.
15
16
5
8
7
62
The Talent Show
1. In the Talent Show, \ of the time was
allowed for singing, and of the time for com-
edy acts.
a. Was more or less time allowed for sing-
ing than for comedy?
b. What part of the time was left for other
kinds of acts?
2. To make posters for the show, Joe and
Tom used some large sheets of heavy paper,
all the same size. Joe cut his sheets into fourths;
Tom cut his into halves.
a. Whose posters were larger?
b. Joe cut up 2 large sheets, and Tom cut
up 3. How many posters was that?
3. There were 12 acts in the show, and 24
contestants. What was the average number of
contestants in an act?
4. The Talent Show began at 7:45 p.m. and
ended at quarter past 9. How long did it last?
5. From Ex. 3 and 4, find the average length
of an act.
6. Tickets for the show cost 40^ for adults
and 25^ for children. The total sales were
200 tickets for adults and 100 tickets for chil-
dren. How much money was received for the
tickets?
Mumhe/i and
1. There are many ways of writing 100 with-
out using zero. Explain Ex. a and b.
a. 99f =100 because
| b. 98 + 1 + J + f J = 100 because
2. Write 78, using only the figure 7, but
repeating it as often as you wish.
3. Write nineteen, using four 9’s.
4. Write two, using four 7’s.
5. Years ago there was a town in which J
of all the people were one-legged, and J of all
the others went barefoot. How many shoes
were needed?
63
Adding and Subtracting Unlike- Fractions
1. The sum of 3 pigs and 4 goats can be
neither pigs nor goats. But 3 pigs and 4 goats
are 7 animals.
You cannot add or subtract things unless
they are
You cannot add or subtract fractions unless
they have the same
1
2
1
2
1
J
l
1
l
5
3
1
1
J.
1
1
1
6
6
6
6
6
6
Use the diagram for help in Ex. 2-5.
2. A fractional unit that measures both y
and 7T is ; so a common denominator for
y and \ is
[ Common denominator not present]
5. In the diagram, mark the bar for ^ s to
show y’s. Mark the bar for y’s to show y^-’s.
The diagram now shows that a common frac-
tional unit for § and § is
6. A short way to find a common denomi-
nator for § and f is to
(add; multiply)
together the two given denominators, 3 and
The common denominator is
2 3
3 + 4 = ~f2~+l2~ = IT =
7. To add J and yq, you could find a com-
mon denominator by
together the given denominators. Then the
common denominator would be
Change J and yo t0 fractions with the frac-
tional unit yo and add.
8. It is better to use the smallest common
denominator. To find the smallest common
denominator for \ an^ A? begin with 10, the
larger of the given denominators.
Does 10 exactly contain 4, the other de-
nominator?
Try 2 X 10, or . Does 20 exactly
contain 4? Is 20 the smallest com-
mon denominator for J and
Add or subtract in Ex. 3 and 4.
To find the smallest common de-
nominator when no given denominator
is a common denominator, multiply the
largest given denominator first by 2, then
by 3, and so on, until you get a number
that exactly contains all the given de-
nominators.
64
a
1.
1
6
2
3
7. 1 2 1
+ 7!
Practice in Adding and Subtracting Fractions
Add or subtract as the sign tells you. In each example, use the
smallest common denominator.
b
2
3
3
8
3
4
1
3
d
+
8
- 3
25
- 1 2
+
3
- 1
6
- 5
9!
+ 81 ~6i
65
Wlto OMUl •— * I (r. 00 ! i- * OOlLn^lW WltO^IOJ LlW Ml h- I I h- ui ho
Rounding Mixed Numbers
6 ft.
X: 5 ft. 2 in. Y: 4 ft. 10 in.
X: ft. Y: ft.
1. On the lines above, write the heights of
X and Y in feet, as mixed numbers.
1 ft. = 12 in., so 1 in. = ft.
2. X’s height comes between 5 ft. and
ft. but nearer to _ __ ft. than to ft.
So, rounded to the nearest whole number, X’s
height is given as ft.
3. To the nearest foot, Y is ft. tall.
4. Both X and Y are about ft. tall.
5. In rounding to 5, you dropped the
fraction because ^ is than
6. In rounding 4f to 5, you counted f as
I, because f is than J.
In rounding a mixed number, when
the fraction is equal to or greater than
y, add 1 to the whole number and drop
the fraction. When the fraction is less
than drop it.
Round to the nearest whole number.
7 3 5 1J3 i 5
‘ • 4 8 16 1 12
8. 14J 5J 34 9|
Estimate the answers by rounding.
9. ft + 2|- + — =
10. 9J - 5ft - =
II. 3|+2f + =___
12. rf + 2J - If
+ - =
13. 1|- — ft H~ 2§
- + =
14. 3* + 2ft + f
+ + =
Is It True? Answer "Yes" or "No"!
1. Are f and f like-fractions?
2. In f, is the numerator 5?
3. Is 3^ a fractional unit?
4. Can ft be reduced?
5. Has | the same value as ft? _
6. Do 4 tablespoons equal \ cup?
7. Is y the fractional unit in ft? .
66
8. Does f mean 5 ^8?
9. Does \ + 4 = = f? -
10. Are there 2 cups in 1 pint? _ _ _
11. Is in lowest terms?
12. Is ft less than ft?
13. Do 6 qt. equal | pk.?
14. Does a leap year have 366 da.?
Look before You Leap!
You should think about a problem before
you try to solve it. There are several ways to
do this.
A. What are you to find?
On the lines below each of problems 1-3,
write what you are to find.
1. Tom had 55^ left after he spent 35^.
How much money did he have at first?
2. Sam earned $1.20 for 2 hours of work.
What was his hourly pay?
3. Celia bought 2 records at 39^ each. How
much did she spend?
B. Decide whether a problem tells about
groups to be combined or a group to be sep-
arated from another group or a group to be
compared with another group.
On the lines below each of problems 4-6,
tell about the groups in this way.
4. Ann’s sister bought a handbag for $3.50
and gloves for $1.35. How much more did
she spend for the bag than for the gloves?
5. Judy’s mother doubled the recipe for some
cookies that required 2 cups of flour. How
much flour did she use?
Think before You Solve!
[Helps in problem-solving ]
6. Jim had 292 bricks for the brick walk he
is making. He has used 189 bricks. How many
bricks does Jim have left?
C. Decide whether to add, subtract, multi-
ply, or divide.
On the lines below problems 7-9, write the
names of the processes that you will use to
solve the problems.
7. Joe allows himself 10 min. to get dressed,
5 min. to clear up his room, and 15 min. for
breakfast. If Joe leaves the house at 7:45 A.M.,
at what time should he get up?
8. The school spent $63.51 for 29 new books.
How much did each book cost?
9. In 5 games a basketball team had the fol-
lowing scores: 59, 73, 77, 64, 82. Find the
average number of points scored per game.
D. It is a good idea to estimate the answer
before you work a problem.
For each of problems 1, 3, 4, 6, 8, and 9,
use round numbers and estimate the answer.
Write the estimated answer at the end of the
line below the problem.
Now go back and solve each problem. Do
your work on a separate paper and write your
answers below.
1
5
2
6
3
7
4
8
9.
67
b. About how much paint would be needed
to give the platform 2 coats of paint?
Last summer, Bill’s grandfather helped Bill
make a house in a big tree.
1. The platform was 8 ft. long and 6 ft. wide.
What was its area?
sq. ft.
2. Grandfather said that 1 gal. of paint covers
about 400 sq. ft.
a. How many square feet would 1 qt. of paint
cover?
3. Bill cut up two planks to make braces for
the platform. Each long brace was 4 ft. long,
and the short braces were each 3 ft. long.
a. Mark the diagram below to show how you
would cut 4 long braces and 2 short braces
from the two 12-foot planks.
i " = — □
i ~i
5. For a hoist to haul up supplies, Bill bought
a pail for 59 £ and 15 ft. of rope at 5^ a foot.
The hoist cost how much?
h 12' H
b. Use color to show what is left after cut-
ting all the braces. What is the total length
of the plank left?
4. One day, Bill and Grandfather each worked
5 hr. Joe helped 2 hr. in the morning, and Ed
and Tom each worked 2 hr. in the afternoon.
This was equal to how many hours of one per-
son’s time?
6. After the first rain, Bill wanted a canvas
roof for the tree house. To allow for the
slant and the eaves, Grandfather said they
needed a piece of canvas 2 ft. longer and 1 ft.
wider than the platform.
a. Mark at the right the
dimensions of the canvas.
68
b. They needed
square feet of canvas.
Testing What You Have Learned
1. Write the fractional unit for each of these
fractions:
5 3 6 7
12 8 7 10
2. Draw a ring around the larger fraction in
each pair below.
1 i. 2 _7_ 1 1 4 JL
53 4 33 12 83 6 93 10
3. Change each fraction to an equal fraction.
[Cumulative Review]
6. Divide. Write any remainder in a fraction
in the quotient.
a. 5JT~92~ b. 8]2T6 c. 4)304
6
10
5
8 ~~ 4
16 ~~
8
10 :
1
9
6
2 ~ 8
12 ~
~4~
9
4. Change to best form.
1 9 _ 6 _
3 — 6 —
16 -
1 8 —
18 _
4 ~
HS
II
=
14 _
16 “
200 _
3 —
5. Add or subtract. Write answers in best
form.
d. 6j5Y0 e. 3)4T7 f. 7)9~05
7. Find the following fractional parts:
1
1
1
1
1
II
00
<4-t
o
h|n
d
___ d. £of 10 =
b. i of 12 =
____ e. y of 14 =
c. | of 23 =
f. | of 15 =
Add or subtract.
8. 23 5
107
+ 692
9. 3,012 10. 6,375
960 8,680
+ 2,1 5 9 +40,09 6
11. 78 2
- 595
12. 1,576
- 807
13. $152.98
-103.79
Multiply. Write all your work here.
14. 4 8 6
X 2 3
15. 5,127 16. $69.75
X 4 7 X 5 8
1.
69
The Junior Choir
[Whole number x fraction ]
Do your work in the space below.
1. In the Junior Choir there are 25 boys
and 15 girls. How many children are there
in the choir?
2. What part of the choir are the 25 boys?
j£_- the 15 girls?
3. It takes f yd. of silk for each tie. How
much silk is needed for 25 ties?
5. Last week the choir rehearsed ^ hr. on
Sunday, fhr. on Thursday, and ljhr. on
Friday. How much time did they rehearse
last week?
4. A collar can be made from | yd. of ma-
terial. For 40 collars, how many yards are
needed?
6. This week the choir has rehearsed f hr.
on each of 3
in all?
days. This is how many hours
Space for Work
Multiply as shown in Ex. 7. Give answers in best form.
7. 4X1 =
4x5
— . — — 04
- 8 - Z«
9i
z2
8. 8 Xf =
9. 10 X § =
10. 4 X =
70
11. 6 X f =
12. 9 X J =
13. 5 X i =
14. 3 X A =
V •
Multiplying a Mixed Number by a Whole Number
many fourths are shown in all?
The diagram shows that 2f = .
2. Box A also shows all the steps in changing
2f to Study the work.
In changing a mixed number to an improper
fraction, how do we know the fractional unit of
the improper fraction?
10. In box B we first change the mixed num-
ber to an fraction. Then
we work just as we do in multiplying any
by a whole number.
11. Box C shows an-
other way to multiply a
mixed number by a whole
C ,3
c
X 5
number.
3? (5 x f)
In box C, how do we
10 (5 X 2)
get the first partial prod-
uct, 3 j?
13|
How do we get the second partial prod-
uct, 10?
In Ex. 12-16, multiply as in box B. In
Ex. 17-19, multiply as in box C.
Change these mixed numbers to improper
fractions. Change the whole-number part in
your head and write your work as in Ex. 3.
1 3. 4i =¥+i = —
4. 6§ =
5. 8J =
j 6. 1\ = -
1 7. 4| =
8. 5^ =
Box B shows one way to multiply a mixed
number by a whole number.
12. 2 X 3§ =
13. 3 X 5§ =
14. 6 X 8f =
15. 8 X 2f =
16. 7X31 =
17. 6^ 18. 9|
X 4 X 4
19. 3§
X 6
77
Envelopes of Fractions — A Game
One rainy day, Grandma
showed Sally and her cousins
Jim and Bill a game with
fractions.
Each player had an en-
velope ruled as shown at the
right, with a number on the
flap.
On a large sheet of brown
paper, Grandma wrote some
numbers and examples. Sally Jim Bill
The game is to copy on the envelope all
the numbers and examples which fit the num-
ber on the flap.
Below are Grandma’s numbers and examples.
Finish Sally’s envelope, and then do Jim’s and
Bill’s.
2i5o
12
4
7-4
lf + lf
'X 1
3 2
6 h- 2
2X1*
li + U
?_5_ 1 _7_
^12 i 12
2 - 1*
5
2
2 Xf
1 -*
i +t
lf+i
2 X 1*
_§_
1 0
1 2
15
The Game of Letters
Grandma showed the children another game.
She gave them the words, “Fractions in Arith-
metic,” and told them to use the letters to spell
as many words as possible. She made these
rules:
1. Use each letter only once. Cross out each
letter as you use it.
2. You must not make more than one
2-letter word.
3. No 1 -letter words are allowed.
4. Count your score as follows:
Give yourself 1 for each letter used.
Give yourself 5 for each word made.
Subtract 2 for each letter not used.
Grandma said, “You can use all the letters
if you try!”
Now you play the game of letters.
metimemow® oo oom>o— ♦
Words
Score
Count for letters used:
Count for words made: +
Sum:
Subtract for letters not used:
Score:
72
Multiplying a Whole Number by a Fraction
1. Draw lines to divide the 8 squares into
4 equal groups.
a. Put X on 5 of the squares.
b. There are in each of the 4 equal
parts of 8.
c. Jof 8 = ____
2. Draw a ring around § of the dots above.
a. How many dots are inside the ring?
b. f of 10 = ____
c. Does the diagram show that the result in
the box below is correct?
c t r\ 3X 10 30 ^
f of 10 = X 10 = — y- = J = 6
3. 10 Xf = -----
a. Is this result the same as the product of
§ X 10 in the box above?
b. You know that you can reverse factors
like 2x3; that is, 2x3= , and
3X2=
The box shows that you can multiply § X 10
just as you multiply 10 X f.
J X 8 has the same value as 8 X
c. We usually read J X 8 as of 8,” and
f X 10 as “ ”
4. Draw a diagram below to show how many
§ of 6 are.
f of 6 =
Find the products as in the box. Give answers
in best form.
5. f X 24 =
6. xo X 105 =
7. f X 4 =
8. i X 29 =
9. | X 27 =
10. | x 84 =
11. f X 18 =
12. * X 20 =
Below each problem, write its solution.
13. Ann bought f yd. of ribbon. How many
inches was that?
14. Jack ran \ mile. How many yards did
he run? (1 mi. = 1,760 yd.)
15. It takes Sam’s father f hr. to go from
his house to his office. How many minutes is
that?
16. On the day of the bad storm, ^ of Joe’s
class of 42 pupils were absent. How many
pupils were absent?
73
Multiplying a Fraction by a Fraction
1. Mother said, “There’s half an apple pie in
the pantry. You three boys may
divide it equally among you.”
What part of the whole pie did
each boy get?
a. The half pie is divided into 3 equal pieces,
so each boy gets of J pie.
b. In the whole pie there are halves,
so in the whole pie there are 2 X 3, or ,
pieces of the size that each boy gets.
c. Each boy gets of the whole pie.
ij of \ pie = J of the whole pie.
2. Mark or color the diagram above to show
J’s. Then mark § of f.
| of f of the bar = of the bar.
3. Sam’s father spends f hr. in going from his
home to his office. (See Ex. 15, page 73.) For §
of that time he rides on the bus. He walks the
rest of the way. How long is he on the bus?
2V3 2X3 V
3 X 4 = 3*4 = _ = — (h°Ur)
Does your answer agree with the diagram
in Ex. 2 above?
Multiply. Give answers in best form.
4. ixM
5. f X | -
6- | X | =
7. i X i =
74
8- f X * =
9. f- X | -
10. yq X To =
11- TO X t =
12. t xH =
Find answers for problems 13 and 14 by
doing your work in the spaces provided. Be
sure your answers are in best form.
13. Ann’s mother had § yd. of red calico.
She used J of it for pockets on an apron. How
many inches did she use for pockets?
a. Change § yd. to inches and find \ of
the number of inches.
b. Find ^ of § yd., and change to inches.
c. Are your answers in a and b the same?
14. Kate’s sister planned to make some
pickles. The recipe called for 12 cucumbers,
but she had only 8 cucumbers.
a. What fraction of each of the other in-
gredients should she use if she uses only 8 cu-
cumbers?
b. The recipe for 12 cucumbers requires
f cup of salt. How much salt should she use
for 8 cucumbers?
Multiplying a Mixed Number by a Fraction
1. The pail that Bill used as a hoist for
his tree house (page 68) held 2\ gal. Grand-
father told Bill to fill it only f full of water,
to avoid spilling.
a. Draw a line to
show the water line
when the pail is f
full.
b. f of the pail is
gallons. (See
the gallon line.)
To find f of 2\ with figures, you can
change the mixed number to an improper
fraction and multiply as you do with any
fractions.
2i = f + i = f
4 v5 =iX5 = 20
5 A 2 5X2 10
c. Does the diagram show that the multipli-
cation in the box is correct?
Multiply. Change the mixed numbers to
improper fractions in your head if you can.
Give answers in best form.
2. f X 10tt =
3. } X 5| =
4. | X 6| =
3* IT) X 4f =
6. 3 X 5f =
3. i X 3y =
9. f X 3£ =
10. f X 5f =
11. A X 9f =
12. §X7i =
13. i X 3 \ =
14. | X A =
In working with fractions you need to do
a lot of multiplying and dividing quickly and
accurately. Here is a good chance to get some
practice with numbers you use often.
Multiply or divide, as indicated. Try to
think the work and write just the answers.
a
b
15.
2X3= ____
8X4= ____
16.
9 _
3 —
5X7= ____
17.
4X6= ____
10 -5 = ___
18.
14 -2 = ____
9 X 3 =____
19.
7X8 =____
1 6 _
4 —
20.
11
CM
•I*
00
v— H
7 X 3 =___.
21.
15 _
3 —
15 _
5 —
22.
6X9= ____
12 -5- 4 = ___
23.
16. _
2 —
9X2=
24.
1
II
•1-
O
<N
18 -3 =
75
7. f X 5f =
Multiplying by a Mixed Number
1. In all the boxes., the
is a mixed number.
2. Boxes A and B show two ways of multi-
plying a ------ number by a
mixed number. In box , the mixed num-
ber is not changed before multiplying.
3. In boxes B, C, and D, what has been done
to the multiplier, 3J?
4. Which box shows how to multiply
a. a fraction by a mixed number?
b. a mixed number by a mixed number?
In problems 5 and 6, you must multiply
with the same kinds of numbers used in the
boxes. For each problem, write the letter of the
box that shows the example you would follow.
Then write the multiplication example for the
problem. Do not solve yet.
5. Bill travels § mi. to school. Ed travels
3^ times as far. Ed travels how many miles?
Box Multiply X
6. Bill and Ed rode for 2\ hr. on their bi-
cycles at an average rate of 9^ mi. an hour.
How many miles did they ride?
Now solve Ex. 5 and 6 in the boxes below,
and draw rings around the answers.
Ex. 5
Ex. 6
Multiply. Give answers in best form.
H xf =
8.
n x it I
05. y 4 _
S' 9
Box Multiply
76
X
Cancellation with Fractions
Multiply: f X lf
4 3
Long way: - X -
4X3
9X2
12
18
2 1
it 2
Short way: ^ X ^ — -
3 1
1. In the long way in the box above, the
product y§ is reduced to lowest terms by divid-
ing both numerator and denominator by
When we divide both the numerator and
the denominator of a fraction by the same
number, are we changing the value of the
fraction?
2. Study the short way in the box. Often
we can make our work shorter by dividing
both a numerator and a denominator by the
same number before multiplying the fractions.
We divide both the 4 and the 2 by ,
and we divide both the 3 and the by
3. The product of the numbers 2 and 3, by
; which we divided in Ex. 2, is
The number by which we divide in reducing
If to lowest terms is
4. Show how you could reduce y§ in two
steps. Divide both terms first by 2, and then
by 3.
12 _
18 —
Cancellation saves work and time, and most
people think it is fun, too. But be sure to
divide both a numerator and a denominator by
the same number. That’s part of the Golden
Rule of Fractions!
Multiply. Use cancellation
sible. Test products mentally.
when it is pos
a
b
5- |xf =- —
1 -
10 ^ 8
6. | X | = — -
16 X 9
7. fxU
1 xz _
2 X 8
8- !xi “ —
A x5 _
15 X 6
l vi -
5 X 2
10. § x| = _
5 x A _
6 X 10
n. I x| =
4 X 5 “ — -
12. fx J-l
1 y io _
10 ^ 21
13. £x?-„
9 2
16 X 3 = -
14. fxf---
3 v Id _
8 X is
!5. f2xf4 =--
1 _
8 X 3
16. *|t=— -
7 X 5
17. |.x| =_ —
t8.1_
10 ^ 12
18. |xf5=--
2 X 3 i
77
Cancellation with More than Two Numbers
Boxes A and B show examples in which
more than two fractions, mixed numbers, and
whole numbers are multiplied together.
In examples of this kind, you must change
any mixed number to an improper fraction. It
is not necessary to change a whole number to a
fraction, but it is a good idea to do so, to avoid
mistakes in cancellation.
1. In box A, the mixed number
is changed to the equal fraction
2. In box B, the 9 is changed to
3. In box B, the 3J is changed to
f X J X 2i = ?
0 v7 =12<J _ 14
0^3 3X3 9
= U
1 9
B
9X§X3ixf=?
3 2 1
I v2 -y* vf
1 a2 a g xg
111
12
4. The cancellation in box B shows that w
divided by what numbers?
Multiply. Use cancellation when possible. Give your answers in
best form.
5 ^ x3 x2 =
h. 6 X 4 X 5
6 ? 21 8 =
3 X 32 X 9
o i 9 5
7. J5 X 2 x 3 X 3
q ^ 6 5 8
4 X 7 X 6 X 15
3 15 3
9- To X 2 X 9 X 4
10. 8 X 16 X 9
12. 9 X A X If =
13. 18x3x3x4 =
14. 3 X 25 X i X H =
15. Tony wanted to write 6 examples in a column on his Space for work
paper. If he needed 1J in. of space for each example, how high
would the column be?
78
Testing a Product
You can test a product easily by noticing
vhether the multiplier is more or less than 1.
Study Ex. a-c in the box below.
a. 4 X 3 — 3 — I3
K J, y 7 __ 7
**• 2 A 8 — 16
1. In Ex. a, the multiplier, 4, is
|han 1. The product, , is
han the multiplicand,
{more; less )
{more; less )
[Size of product related to size of multiplier ]
2. In Ex. b, the multiplier, , is
than 1. The product, ,
is than the multiplicand,
3. In Ex. c, the multiplier is 1. The prod-
uct is the same as the .
If the multiplier is less than 1, the
product will be less than the multipli-
cand.
If the multiplier is more than 1, the
product will be more than the multipli-
cand.
Finish Ex. 5-10 below. Work as in Ex. 4.
Compare Multiplier with 1
Compare Product with Multiplicand
6.
8.
f X7
--7--- IS -
_ Jm.
- than 1.
Product will be _ .
J&M/
_ than . J.
3 Xi
is
than 1.
Product will be
- than _ _ _ .
3 y i
4 A 4
is
_ than 1.
Product will be
than __ .
2J X 1
is
than 1.
Product will be _ _
_ _ than
t X.2
is
_ than 1.
Product will be .
than
1 xA
ii x 14
Product will be
is
^ than 1.
Product will be
than
Now find products for Ex. 4-10
, and write your answers below.
4
Ex. 5
Ex. 6
__ Ex. 7 _
Ex. 8
Ex. 9
Ex. 10
Multiply in Ex. 11-18.
Test the products in the way explained above.
2 y ai _
3 A “
15. I5 X 1&2 —
4^
X
II
16. i X f =
Hr X2i
=
17. 4i X 5i =
=
18. f X 2| =
79
Rounding, Estimating, Testing
Do you remember this rule?
To round a mixed number to the
nearest whole number: (a) if the frac-
tion is equal to or greater than add
1 to the whole number and drop the frac-
tion; (b) if the fraction is less than
drop it.
See how the numbers below are rounded.
1| becomes 2. 2| becomes 2.
2\ becomes 3. 2§ becomes 3.
3J becomes 3.
In Ex. 1-10 below, round each mixed num-
ber to the nearest whole number.
1. 2. 3.
4. 5. 6. 7. 8. 9. 10.
Mixed number ....
10i
Q3
4*
6A
8|
3-2-
J1 0
7 1.5
' 1 6
->7
A15.
^32
02
Z3
Rounded number . . .
You have to know when to use the rule
above and when to estimate differently. Some-
times it is better not to follow the rule! Let’s see
if you have good judgment.
11. Sue said, “We live about 3 mi. from
town.” The exact distance is 2^- mi. Was Sue’s
estimate a good one?
14. Bill made this estimate for 2^% X 8J-:
“2 X 8 = 16, and 3 X 9 = 27, so the answer
lies between 16 and 27.”
a. Explain Bill’s estimate.
12. Dee’s father measured a boundary line
and found it to be 1,316^ ft. Dee asked,
“Would you call it 1,317 ft.?”
“Yes,” her father said. “A legal descrip-
tion might be T,317 ft. more or less’; but for
some purposes I might round 1,316 yq ft. either
to 1,300 ft. or to i mi.”
Explain.
13. Don’s father said, “Cut me six pieces of
molding 21J inches long.” Should Don round
21 J in. to the nearest inch? Explain.
b. The exact answer is Does this
agree with Bill’s estimate?
15. Betsy needed 25^ in. of ribbon for a hat
band. She bought fyd. Did Betsy make a
reasonable estimate?
Explain.
16. The telephone “Time Service” gave the
correct time as “8:47 and 50 seconds.”
a. Mrs. Huey set her clock at quarter of 9.
Was the clock fast or slow then?
b. To the nearest minute, the correct time
was minutes before
80
Sometimes TV actors read their “lines”
from cue cards, which are not seen on the
TV screen. These cards are rectangles of card-
board with words in large print.
The diagram below shows a cue card and
this workbook drawn to the same scale.
Scale Drawing and Television
[ Meaning of scale ]
After Ed saw a dog show on TV, he looked
up “Collie” in his dictionary. He found a picture
that looked like this.
5. The scale is
6. To find about
how long the collie is,
you
the length of the drawing by
7. The length of the drawing is
So the real collie is about in., or
long.
Scale: ^ = 1"
Workbook
1. The scale tells that in. on the drawing
represents in. on the cue card. Then 1 in.
on the drawing represents in. on the card.
2. The width of the cue card in the draw-
ing is yq of the width of the actual card; so
width of card = X width of drawing.
3. Find the dimensions of the card.
Drawing Card
Width: in. _ „ _ in.
Height: in. in.
4. By thinking how many workbooks can fit
on one card, you can estimate that the card
is about how many times as large as the work-
book?
K -Sca±-7 XbZ- — -a
Rough-haired Collie
m.
ft..
8. The height of dogs is measured from
the withers (shoulders) in a straight line to
the ground, as shown by the dashed line in the
drawing.
The height in the drawing is inch,
so the real collie’s height is about in.,
or ft.
Scale models can sometimes be used in set-
tings for moving pictures and TV productions.
Skillful use of the camera makes them appear
to be full size.
9. One scene in a film shown on TV rep-
resented an earthquake. The buildings were
models to the scale 1 ft. = 12 ft. Find the
width and height of the model representing a
house 34 ft. wide and 20 ft. high.
Width: Height:
10. The model of a factory building was
8f ft. long. How long was the real build-
ing that it represented? (Use the scale in Ex. 9.)
11. A church steeple 88 ft. high was how
high in the scale model?
87
Making and Reading Scale Drawings
1. Make three scale drawings of a line that
is 8 in. long. Use these scales:
Scale, \.
Scale, tV
Scale, J.
Mouse
[Using a scale ]
Cat (A)
2. In Ex. 1, which of the scales made the
largest drawing? the smallest draw-
ing? The larger the scale is, the
the scale drawing is.
3. If you draw a picture of a bird to the
scale J, and then draw a picture of the same bird
to the scale J, will the first drawing be larger
or smaller than the second drawing?
4. If you want to show a 10-inch line on
7. If you had never seen either a cat or
a mouse, how could you tell from these draw-
ings which animal is larger?
The real mouse is times as large as the
drawing; the real cat is times as large.
8. This could be a picture of Susan’s doll’s
teacup or of Susan’s mother’s cup, depending
on the of the
drawing.
a drawing to the scale 1 in. = 4 in ., you think ,
Is it a doll’s cup if the scale is J?
“My line must be as many inches long as
there are 4’s in 10, or inches.”
5. To the scale 1 in. = 4 in ., a line 6 feet
long would be in. long in the drawing.
Explain.
6. Last summer, at his grandfather’s farm,
Richard collected snakes. Finish the list of
lengths below. Then draw lines to represent
these lengths. Use the scale 1 in. =16 in.
a. Striped adder, 2\ ft., or in.
b. Garter snake, 1 J ft., or in.
c. Black snake, 3f ft., or in.
a.
if the scale is -J?
9. Scale drawings are not always smaller
than the things which they rep-
resent. Look at the drawing
of a mosquito, at the right.
Is it larger or smaller than life
size?
Scale: f
10. The mosquito was drawn to what scale?
You know that a scale of \ means that
1" in the drawing stands for " in the real
object. Then a scale of \ means that 2" in the
drawing stand for " in the real object.
11. In the drawing the wingspread of the
b.
c.
82
mosquito is ". The wingspread of the real
mosquito is j of f", or ".
Making and Reading Graphs
1. Uncle Pete gave Joe some out-of-print
United States stamps for his collection. Joe
made a record when he sorted the stamps.
a. In this record, / means stamp.
b. Complete the table.
Stamps 1^ 2 <ji 3 i 4^ 5^
Tally ///// ///// ///// // /////
/ ///// ///
Number
Total number of stamps:
2. Uncle Pete showed Joe how to draw a
picture graph. Finish this graph:
r»iiiiiiini
Key:
|~~j = 1 stamp
V
Joe’s Stamps from Uncle Pete
3. The key of the graph shows that each
picture stamp stands for stamp.
Joe thought that a picture graph is easy to
read, but it takes a lot of time to draw. So
Uncle Pete made this bar graph i
0 2 4 6 8 10 12 14 16 18 20
Number of stamps
Joe’s Stamps from Uncle Pete
4. The graph above is called a horizontal-
bar graph because i
t 3*|
s *i
54 0
[ Picture graph and bar graph ]
5. Finish the horizontal-bar graph.
6. Joe thought that this bar graph is some-
thing like the picture graph. See if you can
tell why Joe thought so.
To show how many stamps he has from dif-
ferent countries, Joe made the vertical-bar
graph below.
Joe’s Stamps from Different Countries
7. In Joe’s graph, each small space on the
number scale represents stamps. Why
didn’t Joe number all the dividing lines?
8. Find how many stamps Joe has from each
country. Write the number after the name of
the country in the list below.
France - India
Canada Mexico
U.S.A. Brazil
Italy
83
Reading Maps and Graphs
1. The picture above is a scale drawing, or
map, of Lakeside Camp. Measure from center
to center of the dots on the map to find the dis-
tances listed below. Then use the scale to find
the real distances.
Distance
Map Real
a. Main tent to pier
b. Cook tent to main tent
c. Woodpile to cook tent
d. Tent A to pier
e. Tent A to shore __
Distance
Map Real
f. Tent A to main tent
g. Tent B to pier
h. Tent B to main tent
i. Tent B to shore
j. Tent B to Tent C
k. Tent B to cook tent
l. Tent C to shore
m. Tent C to cook tent
2. Graph A is a
graph. Each picture of a boat
means sailboats.
3. How many boats had
a. red sails?
b. blue sails?
c. black sails?
d. white sails?
84
Flowers Sold on Monday
4. Graph B is a vertical- graph.
5. On the scale, 0, 1, 2, 3, and 4 mean
6. Each small space on the scale stands for what fraction of
a dozen?
7. How many daffodils were sold?
8. Graph C shows that a #3 can holds cups.
9. Each small space on the scale stands for cup.
10. A #2 can contains cups.
Drawing
Flowers Sold on Monday
Daffodils .
. . 2\ doz.
Tulips . .
. . 3 doz.
Roses . . .
. . 4 doz.
On page 84, a vertical-bar graph was used
to show the facts in the table below.
The same facts
can be shown also
in a picture graph
or a horizontal-bar
graph.
1. In the picture graph below, the key says
that stands for 1 doz. flowers. Then
can be used to stand for doz. flowers.
2. Finish this picture graph:
,
Key=^= 1 doz-
Daffodils
$
Tulips
Roses
Flowers Sold on Monday
3. In the horizontal-bar graph that has been
started below, each small space on the number
scale represents dozen flowers.
4. Finish the bar graph below. Use the num-
ber scale to help you make each bar the correct
length.
Daffodils
Tulips
E
2 3
Dozens
Flowers Sold on Monday
5. In Ex. 2, the number of tulip pictures
is of the rose pictures. Are 3 dozen J as
many as 4 dozen?
Graphs
[ Picture graph and bar graph ]
6. In your bar graph (Ex. 4), the tulips’ bar
is as long as the bar for roses.
7. Suppose you made the bar graph to any
other scale, what would be the relation between
the bar for tulips and the bar for roses?
8. On the squared paper below, draw a graph
for each set of facts below.
Make one a picture graph and the other a
bar graph. Show the scale or key. Label the
bars or rows of pictures, and give each graph
a title. A graph should show all the facts in
the table.
a. Eggs Laid by Ed’s Hens Last Week
Sun. Mon. Tues. Wed. Thurs. Fri. Sat.
10 12 14 10 13 15 9
b. Fish Caught on Saturday
Bill Tom Ed Sam Joe
3 5 2 2 4
85
Add? Subtract? Multiply? Divide?
For each missing word, write “add” or “subtract” or “multiply”
or “divide,” whichever makes the statement true.
Then write and work an example (or two examples if necessary) to
illustrate each statement. It may help if you think of an example before
you complete the statement. Draw a diagram when you think that one
is needed. Number each example like its statement.
Illustrations
1. To change an improper fraction to a whole number
or a mixed number, you the numerator by
the denominator.
2. To find the area of a rectangle, you think of the num-
ber of area units in one row along the long side; then you
this number by the number of rows.
3. To find how many there are in one of the equal parts of
a group, you
4. To change from larger units of measure to smaller units,
you
5. You cannot or fractions un-
less they have a common denominator.
6. To change from smaller units of measure to larger units,
you
7. You can or both
terms of a fraction by the same number without changing
the value of the fraction.
8. To find a difference or a remainder,
9. To find a total, either or .
10. To find how many equal small groups are contained in a
larger group, you
86
Do You Remember?
1. Suppose this is a scale drawing of your
shoe. Measure the length of the drawing; then
measure your shoe to the nearest inch. Find
the scale of the drawing.
Length of drawing: in.
Length of your shoe: About in.
Scale of drawing: About 1 in. = in.
2. On a map to the scale 1 in. = 50 mi., a
line 2yq in. long represents mi.
3. In a drawing to the scale y, a line 7 in.
long represents in. on the real object.
1 pt.
i®
W- 1 cup
1 qt.
»f
if
1 gal.
1 PW
11
if
1®
®1
11
®f
Liquid Measures
4. From the graph above, supply the num-
bers missing in these tables:
a b
lpt. =
cups
1 qt. =
- Pt*
1 qt. = _
_ cups
1 gal. =
- qt-
1 gal. =
cups
1 gal. =
pt.
5. For each fact in Table , you can just
count the pictures in one row of the graph. For
each fact in Table , you compare the totals
for two rows.
[Review: Scale drawing and graphs]
6. Draw a horizontal-bar graph below to show
the facts in Table a, Ex. 4.
To help you choose the scale, think :
a. The longest bar in the graph will stand
for how many cups?
b. There are squares across the
paper from left to right. Then can you let
one small space on your number scale stand
for 1 cup?
7. On the squared paper below, draw a ver-
tical-bar graph to show these heights:
Father, 5 ft. 9 in. Tommy, 4 ft. 9 in.
Mother, 5 ft. 6 in. Sally, 3 ft. 6 in.
Bill, 5 ft. 3 in. Dicky, 2 ft.
Helper. Let 1 square = 3 in., and mark
only each foot on the scale. Make the bars
2 squares wide and 2 squares apart. To label
each bar, write the person’s name on it. Don’t
forget to give the graph a title!
87
Testing What You Have Learned
1. Round to the nearest whole number:
a. 6§ c. 2\ e. 1
b. 8j d.£ f. 1*
2. a. y§X 32 =
b. 2f X 3* =
3. At 50^ a pound, how much does 1^ lb.
of ground meat cost?
4. To the scale \in\ = 1 ft ., how long is
the line that shows 15 ft.?
5. Round to the nearest hundred:
a. 5,280 c. 43,560
b. 144 d. 1,092
6. Change these measures as directed:
a. 3* ft. =
in.
d. 2 lb. = _
oz.
b. 1J bu. =
- Pk-
e. 18 in. =
ft.
c. 5 pt. = .
- qt.
f. 45 in. =
- yd.
7. In the
example
f X 8 = 6,
the prod-
uct is , the multiplier is , the multi-
plicand is , and the factors of the product
are and .
8. Show how to tell which fraction is larger,
2 nr 3
3 UI 4*
9. Draw a ring around the larger fraction in
each pair below.
~ i ix
A 3
7
a* 2 16
Cl. 4
10
b. i i
0 5
e. 8
2
3
88
7 5.
12 8
[' Cumulative Review]
10. Turtles sometimes live to be very old. A
turtle that was first seen on the island of Mau-
ritius in 1766 was accidentally killed in 1918.
About how many years did it live?
11. The sign on a bargain counter said:
REMNANTS - About 1 yd. each
The measurements of 5
remnants on the counter are
Inches
given in the box.
39
a. The average length was
36
40
in.
38
37
b. Was the sign correct?
c. There are rem-
nants longer than the average,
_ shorter, and
just the average length.
d. On the average, were the remnants more
or less than a yard long?
12. A “disc jockey” invited three judges to
score some new records. For each record,
each of the judges put down a score between 75
points (“fair”) and 100 points (“wonderful”).
One record got scores of 79, 81, and 86 points.
What was its average score?
13. Do what the signs tell you.
a* i T § =
b* f X f =
( I ] 2 _ 3 _
U. -L3 4 —
C.
3.
4
f.
The School Zoo
The following 6th-grade pupils lent their
pets to the School Zoo for a week:
Joe ... hamsters Joan .... canary
Jim . . . mice Sue .... rabbits
The children made up problems 1-6.
For each problem, a. find, in the boxes below,
a diagram and a solution that fit and mark each
of them with the example number; b. write
the answer after the problem. Be sure to label
the answer (ounces, days, teaspoons, and so
on) to show what it means.
i 1. Joe feeds \ ounce of pellets a day to each
hamster. How much do his 3 hamsters eat in
a week?
Ans.
2. In Ex. 1, how soon will a hamster eat
1 lb. of food?
Ans.
[ Dividing a whole number by a fraction ]
3. Joe gives the mother hamster J cup of
milk a day. How long will 1 cup of milk last?
Ans.
4. For the frame for his mouse cage, Jim
cut a board 6 in. wide into f-inch strips. How
many strips did he get from the board?
Ans.
5. When one of her rabbits was sick, Sue
put Jtsp. of medicine in the milk twice a day.
How much medicine did the rabbit have in
3 days?
Ans.
6. Joan gives her canary a heaping teaspoon-
ful of bird seed a day. If that is about § tbsp.,
how long will 6 tbsp. of bird seed last?
Ans.
Ex. __
2 Xi =i
3 X i = li
Ex 3 X i = li
' 7 X H = 10i
Ex.
\ | /
\ /
Ex. _
Ex.
■ ©o ©o Oc?
0o ©e? ©o ©^
Ex
0 12 3 4 5 6
1 I I 1 1 J 1 I-.1 1 I i I 1 I I I 1
t t t t t t t t t
Ex.
1 -f- J •= 4 fourths — j— 1 fourth = 4
Ex.
16 - 4= ^- 4= 32-1 — 32
Ex 6 - | = \4- - | = 24 - 3 = 8
Ex 6-§=4f-§ = 18-2=9
Ex
1 j
> 2
1
l (
1 i
l f
i 1
» <
1
f 1 i
’ !
1 1 1
5 <
i !
> 10 1
I I
1 1
1
2 i:
3 1<
I 1
l 1
1 i
5 16
| |
1 1
1
1
i (
I ;
1
1 1 1
: :
— 1 — — 1 — i
l :
‘ !
‘ 1 j
”t ;
T|
Ex
f , , , i1
2 3
I i i i I
4 5 6
1.1 I I I 1 I I I 1
t t t t t
1 i t
89
Dividing by a Fraction
[ Dividend a whole number ]
A
B
4 H-f = ?
0 12 3 4
1 1— — . — ! 1 1 L_J 1 i J L ! J
4+i-f4.,2.2-6
1. The entire number line (box A) shows
ones, each divided into equal parts;
that is, the line shows thirds in all.
2. When we mark off distances of § on the
number line in box A, we are measuring ^
by §, or dividing 4 by
3. The diagram shows that § is contained
exactly times in 4.
4. In the diagram and in the number work
(box B), the dividend, , is changed to an
equal fraction with the fractional unit ,
so that both dividend and divisor will have
a denominator.
Dividing by a Mixed Number
[ Dividend a whole number ]
B
A i 1 8 3 q 8
4 h- 1J = - + - = 8 -h 3, or or 2§
3. In box A, the remainder, 2 halves, is
what fractional part of the divisor, 3 halves?
So the distance remaining is of the
distance needed to measure another § .
4. In box B, what did we do to both divi-
dend and divisor before we divided?
A
4 4 1J = ?
1. Box A. Why is each 1 divided into J’s?
2. After 1^ is measured twice along the
line in box A, the distance remaining is
how many halves?
5. Box B shows that in ^ there are as many
groups of § as there are 2’s in
6. To divide 9 by f, you first change the 9
to —j~. Then you divide by
7. a. Write the division example shown by
the diagram below: -f- =
b. Now do the division with figures.
8. Mark this line to show 4 ■—
90
Finding Mistakes
Many of these answers are wrong. Cross
out a wrong answer like this: X. Then divide
correctly and check (v) the correct answer.
1. 8 1 1 = ^ 1 1 = 72 | 4 = 18
2. 6 - I =f Hf =6 -2 §3
3. 4-|=¥-t=36-4=9
4. 3 - 1| = f - f = 3 # 3 = 1
; 7 P 3 _ 7x3 _ 21 _ t?
K I • 7 ~ J — rj — J
6. 2 -r- ^ = 4 halves 1 half = 4
7 c . 2 _ 10 . 2 _ 10 _ ?1
* • J • 3 — 2 ' 3 3
8. 6+f=4^-=-f=42+3=14
9. 15 + 3f = ^ + = 60 + 15 = 4
[ Dividing whole number by fraction or mixed number ]
10. 17 3J = ^ + J = 34 + 7 = 4f
11. 4-f=4x|=J#=f = li
12. 10 + | = ^8 + f = 60 + 5 = 12
13. 10 = 1| = ^ + ¥ = 80 = 15 = 51
14. 6 f f = ¥ + f = 54 = 5 = 10f
15. 14 + i = ^ | = 1!2 + 7* 17
16. 5 +1J = §+ §= ¥= 2f
17. 8 + 2§ = J-f + f = 16 + 8 = 2
18. 12 + § = 12 + 4= 3
19. 6 + 2\ = ^ + | = 24 + 9 = 2§
20. 1 & i = f
21. 14 + § = -^ + § = 98 + 5 = 19f
91
A lustukesi ^rUcJzl and
ro
All these puzzles are about fractions. Think
carefully and do not let them fool you.
1. A watermelon weighs j pound more
than f of its weight in pounds. How many
pounds does it weigh?
2. If 1J candies cost l-J^, how much will
14 doz. cost?
3. If a peach weighs J of an ounce more
than a plum that weighs | of an ounce, how
much does the peach weigh?
4. What is the number that becomes 20 when
multiplied by 40?
5. A boat was floating in water 23 ft. deep.
The water came 4 of the way up the side of
the boat. Then the tide rose 2 feet. How far
up the side of the boat did the water come then?
6. After ^ of a piece of cloth was cut off,
there were 10 yd. left in the piece. How many
yards were there at first?
Dividing by a Fraction
1. a. 6g| = ? At the foot of the page,
draw a long line and label it A. Mark line A
to show 6 -i- § .
Helper. You are working with fifths, so
the line should show 6 X 5, or 30, equal parts.
You will have room enough to make each of
these parts \ inch long.
b.6^=-
2. Ex. 1 shows that, to divide a whole num-
ber by a fraction, you change the
number to a which has
the same
as the divisor. Then ,
using just the numerators.
or a Mixed Number
[ Dividend a whole number ]
3. a. 6 -r- 1^ = ? Below line A, draw an-
other number line and label it B. Mark line B
to show 6^-1^.
b. 6 -s- 1£ = -g -j- = — + — | —
4. Ex. 3 shows that in dividing a whole
number by a mixed number you first change
both numbers to fractions.
Then you work just as you do in dividing a
whole number by a
5. To divide a whole number by a fraction
or by a mixed number, you express both divi-
dend and divisor as fractions with the same
92
In the examples below, use either a number line or the common-
denominator method, whichever you like better. If you use a number
line, be sure you know what any remainder means.
6. 3 t- j? =
00
•1-
II
7-7 4f =
15. 9 + 2J =
00
■<1
•1-
►Ww
II
16. 12 h- 2f =
9. 5 # # =
17. 3 h-| =
10. 6 h- 11 =
18. 10 f =
11. 5 +* =
19. 2 -s- i =
12. 15 h- f =
20. 6 h- f =
13. 10 + 3| =
21. 12 -f-f =
22. To check a quotient, you can multiply
of 3, the divisor, and 2, the ,
the _ _ and the
equals the dividend, _
to see if the product equals the _
Check in this way the quotients you found
Helper. Think of 6 + 3 = 2. The product
for Ex. 6-21. Work below the examples.
93
Ratios
1. This picture shows the fish that won Bill
the prize in the fishing contest.
a. The scale J means that 1" on the draw-
ing represents " on the fish.
b. The drawing is " long.
c. Bill’s fish was " long.
2. The scale also means that the drawing
is as long as the fish; or, to put it the
other way, the fish was times as long
as the drawing. The scale in a scale drawing
shows a relationship, or ratio.
3. The biggest fish caught were:
Bill’s, 16"; Tom’s, 12"; Ed’s, 8"
Draw lines to the scale J to represent the
boys’ fish.
0 1 2
Bill’s:
Tom’s:
Ed’s:
4. Compare Bill’s and Tom’s catches:
a. Bill’s fish was times as long as
Tom’s; b. Tom’s was as long as Bill’s.
5. Compare Tom’s and Ed’s catches:
a. Tom’s fish was times as long as
Ed’s fish; b. Ed’s fish was as long as
Tom’s fish.
94
6. Use ratios to compare Bill’s fish with
Ed’s fish in two ways.
7. Sally said, “I got 8 answers right on a
test of 10 questions.”
Write these ratios in best form.
a. Ratio of right answers to total:
b. Ratio of right answers to wrong:
c. Ratio of wrong answers to total:
d. Ratio of wrong answers to right:
8. Now use these ratios in sentences. Label
each sentence with the letter of the ratio that
you used from Ex. 7.
( ) Sally got of the answers right.
( ) She got of the answers wrong.
( ) Her answers were right times as
often as they were wrong.
( ) She got wrong answers as many
times as she got right answers.
9. Did Sally miss 2 questions?
Why isn’t 2 a ratio here?
10. Tom’s father is 40 years old, and Tom
is 12 years old.
a. Tom’s father is times as old as
Tom.
b. Tom is as old as his father.
What Does the Quotient Mean?
[How many times ? What part off]
1. When we compared Bill’s 16-inch fish and Ed’s 8-inch
fish, we found that
a. Bill’s fish was -1* or times, as long as Ed’s.
b. Ed’s fish was y8g-, or 5 as long as Bill’s.
2. 15 — 5 = This means that 15 is times 5.
3. 5 -h 15 = This means that 5 is of 15.
4. 4 - § = — - 1 = = The quo-
tient means that 4 is times §.
5. 4 1 24 =
The quotient means that 4 is times 2\.
6. 4 - 6§ =
; The quotient means that 4 is of 6§.
Numkesi <1>UcJzl
astd PuffileA
1. If 6 cats can eat 6 rats
in 6 minutes, how many cats
will it take to eat 100 rats
in 100 minutes at the same
rate?
2. Joe had 5 sandwiches
and Ted had 3. Sam offered
to pay 40^ if he could share
their lunch. If they all ate
the same amount, and ate all
the sandwiches, how should
the 40 ^ be divided between
Joe and Ted?
■
Joe, Ted, £
You can tell whether the quotient will be a how-many-
times number or a what-part-of number if you notice
whether the dividend is larger or smaller than the divisor.
Complete Ex. 7 and 8 by inserting either “larger” or
“smaller.” Use Ex. 1-6 to help you.
7. The quotient is a how-many-times number (that is, a
number than 1) when the dividend
is than the divisor.
8. The quotient is a what-part-of number (that is, a num-
ber than 1) when the dividend
is than the divisor.
Without dividing in rows 9-13, copy each example in the
box at the right to show what its quotient will mean.
a
b
c
d
9.
25-5
3-2
12-|
23-6
10.
7-28
8-9
9-27
2 — 3y
n.
3 — T
14 15.
A4: ' 16
8-3
15 -20
12.
10 -6i
7 1 1\
14-2*
19 -4
13.
1 . 7
2 • 8
39 - 13
10 - 16
15-8
In Rows 9-13 the Quotient Tells
How What
Many Times Part of
95
Dividing a Fraction by a Whole Number
[ Inverting the divisor ]
A
Show f -5- 2 by a diagram.
(»i r iiii r~ r~i
(2)i til i i'.-i.:T~n:'rrm"n
<»rm i i i mim rrm
j
Complete these statements about box A:
1. Bar (1) shows 5 colored.
2. Bar (2) shows each divided into
2 parts, so each \ of J- is of 1.
3. Bar (3) shows that J of f = ,
because
4. Is the quotient of f -a 2 greater or less
than 1? How could you
know this without dividing?
5. When you divide any proper fraction by
a whole number, the quotient will always be
than 1.
Now look at box B.
6. Notice that the divisor, , can be writ-
ten as the improper fraction f. If f- is inverted
(that is, if numerator and denominator change
places), it becomes
7. To divide f by 2, you can find \ of f.
B
Divide f by 2.
Think: f - 2 = J of f
i of f = i X f, or f X i
Solution: f -5- f = f X j = A |
8. Check the quotient of f -a 2 by mul-
tiplying the divisor and the
to see if you get the
X = ,or
To divide a fraction by a whole num-
ber, write the whole-number divisor as
a fraction, invert it, and multiply.
Divide. Cancel when you can.
9. fJlr9 =
10. i s- 6 =
11. f -5-4 =
12. & -5-6 =
13. f -5- 8 I
14. | - 4 =
15. f - 2 =
16. § -3 =
17. | - 10 =
18. | - 4 =
19. A - 6 =
2 X 8) 0r 8 X
96
Dividing a Mixed Number by a Whole Number
1. In box A, the dividend is . and
the divisor is . Will the quotient be
a how-many-times number or a what-part-of
number?
Why?
2. In box A, where does the § come from?
. Finish the work in box A.
[ Quotient : how many times; what part of ]
8. Five boys wanted to divide 4f pounds
of nuts into equal shares. Find each boy’s
share.
a. In this example, we are dividing 4§ lb.
into equal parts. That is, we are finding
how many there are in each f of
b. Write and work the division example in
the space below.
Divide in Ex. 9-16. Use cancellation when
you can.
9. 2\ -s- 3 =
10. 1* -s- 4 =
3. The answer in box A shows that If is
what part of 6?
11. 5J -r- 4 =
12. 6f 9 =
4. In box B, the quotient will be a -
- number because the
dividend is than the divisor.
5. Finish the work in box B.
6. Box B shows that 3f is times 2.
7. To divide a mixed number by a whole
number, you first change the mixed number
to an fraction. Then
you work just as you do when you divide a
by a number.
13. 3§ - 4 =
14. 3f - 8 =
15. 3f - 3 =
16. 2f -s- 3 =
97
Alice in Wonderland
In a TV production of the story of Alice in
Wonderland , Alice had to appear very tall after
she ate the cake, and very short when she held
the White Rabbit’s glove.
Since the audience would judge Alice’s height
by the ratio of her height to the height of the
furniture, the effect was accomplished by using
special furniture.
1. The actress who played “ Alice” was 5 ft.
tall. Was she twice as tall as an ordinary table,
2\ ft. high?
2. In order to make “Alice” appear very
tall, would the special table have to be higher
or lower than an ordinary table?
3. When the special table was 10 ft. high,
“Alice” was still 5 ft. tall. She was only
as high as the table. Since the table appeared
to be an ordinary table, 2\ ft. high, how tall
did Alice then seem to be in the picture?
[ Problems with fractions ]
Work the following problems and write the
answers on the lines:
4. The Mock Turtle’s dance took only
2\ min. on the TV screen, but the actors
spent 3 hr. rehearsing it. The rehearsal time
was how many times as long as the screen
time?
5. During the 90-minute TV production,
there were 4 commercials, each of which lasted
If min. The time used for commercials was
what part of the total time?
6. Joan and three friends watched Alice in
Wonderland together. While they watched the
program, they ate f of a 6-ounce package of
nuts. How many ounces was that?
7. For supper, after the TV program, Joan
made cocoa with evaporated milk. She used
If c. of evaporated milk and added f that
much water. How many tablespoonfuls of water
did Joan use? (1 cup =16 tablespoons)
When Alice grew to be 9 ft. tall after eating
the cake, she became confused about arith-
metic. You don’t need to be confused about
solving problems if you remember what you
have learned about fractions.
8. During a game, Joan asked, “If f mile
is marked off into 6 equal distances, each dis-
tance is what part of a mile?” What is the
correct answer?
98
Dividing a Fraction by a Fraction
You can divide a fraction by a fraction by
using the common-denominator method. To do
this, you change the fractions so that they have
a common fractional unit; then you divide the
numerator of the dividend by the numerator of
the divisor.
Finish Ex. 1-4.
9_ 3 _ 9_ ^
16 ' 8 ~ 16 1 16
[' Quotients more than 1 and less than 1~\
7. Ex. b in the box means that § is § as
large as . Why is the quotient a what-
part-of number?
Work Ex. 8 and 9 by inverting the divisor
and multiplying.
« _9 i_ 3. _
°* 16 * 8 —
3.
2 3
3 * 4 _ 12
12
4.
1 ^ 1 =
6 ' 8 24
l _
8 —
5. To divide a fraction by a fraction, you
can divide using just the numerators only when
10. Compare Ex. 8 and 9 with Ex. 2 and
4. Which method of solution seems easier?
the fractions have the same
In dividing a fraction by a fraction, you can
ialways change both dividend and divisor to
fractions with a common denominator; but it
is usually quicker and easier to divide by in-
verting the divisor and multiplying, as you
have done in dividing a fraction by a whole
number.
The box shows Ex. 1 and 3 worked by the
inversion method.
6. Ex. a means that f is times ,
In Ex. 11-16, use the method that seems
easier. Be sure to give your answers in best
form.
12.
5_
6
7 _
12 —
13.
l.
3
1. _
2 —
1 _
6 —
or that there are one thirds in
The quotient, 2J, is a how-many-times number
because the dividend, , is
than the divisor,
15.
8 2 _
9 *3
16.
7
12
3. _
8 —
99
Dividing by a Mixed Number
[ Dividend a fraction or a mixed number ]
A
B
c
5 . <2 _ 5 .20
6 * u3 ~ 6 • 3
4i ^ 52 _ 9 2,7
^2 • J5 — 2 * 5
. -2 3 _ 20. . 15
U3 • ~ 3 ~ 4
1 1
1
4
$ v 2 i
0 v 5 5
2€T 4 16
If X20 8
2 6
3 X 15 9
2 4
3
3
You know how to divide by a fraction. Be-
cause any mixed number can be changed to
an improper fraction, you also know how to
divide by a mixed number.
1. In box A, the dividend, f, is a fraction;
the divisor is a number;
the quotient is a proper , that is,
a number than 1.
2. The quotient in box A shows that f is
of 6§. Why is the quotient of a proper
fraction divided by a mixed number always
less than 1?
7. * - 3^ =
8. 1* 3§- -
Q 3§i_ 11. _
4 • i4 —
10. 3f -s- 2\ =
11. 1* - 3§ =
12. | -s- If =
13. 2f =
14. If - 2f =
3. In which of boxes B and C is the quo-
tient a how-many-times number?
Why?
15. f -r- 2^0 =
16. 3f - 4f =
17. 2\-r 2\ =
4. Do boxes A, B, and C show the common-
denominator method of dividing fractions or
the inversion method?
18. f -r If =
19. 7f - 9f =
Divide in Ex. 5-22. Cancel when you can.
_9 •_ —
10 • A3 —
20. 3f + 4$ =
21. To + If =
6. 3§ 4- lOf =
700
22. 7^- 4- 12£ =
cDj-q
Division with Fractions
1. Judy and Carol are making fruit punch
■or a party. The recipe calls for 1 J tbsp. of
Dunch mix to each glass of ice water. How
nany glasses of punch can they make from a
Dottle of mix containing 24 tablespoonfuls?
Helper. To find the answer, you divide
___ by First you change 1^ to
;he improper fraction
Write and work the example in the space
uelow.
They can make glasses of punch.
Divide. Cancel when you can. Show all your work.
2.
16
. 1 _
• 8 —
13.
2 -
- —
Z2 ~
3.
12
x3
-5 =
14.
5i
_i_ 2. —
* 3 —
4.
20
. 4
• 5
15.
1 .
2 •
-li =
5.
12
•4.1 -
16.
Ol
Z2
-5- 10 =
6.
3i
. 1 _
• 10 —
17.
3 .
8 •
-6 =
7.
2 .
3 *
4 _
' 5 —
18.
4 .
5 •
2 _
' 3 —
B.
+ =
19.
05
Z8
13 _
. 14 —
9.
8§ -
-4 =
20.
00
Wit-1
. 5 _
* 6 —
LO.
5 .
6 *
5 _
9 ~
21.
6 -
2
" 3 —
LI.
1
9 !
-8 =
22.
7
12
.A. 05 _
. z8 —
b.
3 .
4 •
1 _
2 —
23.
->3
-5 =
101
Relationships in M. and D.
Study these relationships:
(1) Factor X factor = product
V — ^2 X*3 = 6^
(2) Product — one factor = other factor
6^2=3
6-3=2
For the division 6 — 3 =2:
(3) Quotient X divisor = dividend
V >2 X*3 = 6* 1
(4) Dividend — divisor = quotient
6-3=2
(5) Dividend — quotient = divisor
6-2=3
1. From statement (2) in box A, you see
that we can find either of two factors of a
product by dividing the by
the that we know.
2. From (3) you see that the quotient and
the divisor are both of the
dividend.
3. From (4) and (5) you see that if we
divide the dividend by the divisor, we get
[ Finding n, the missing number ]
B
Find n when
a. n is the product of two factors.
9Xf=n;n=9X§.
n =
b. n is one of two factors of a product.
c. n is the dividend.
d. n is the quotient.
e. n is the divisor.
-9.
n
. 2
• 3-
n
xf.
n
. 2
• 3*
n
-9.
n
the ; and if we divide the divi-
dend by the quotient, we get the
In each case, we are dividing the dividend
by one of its to get the
other
The relationships given in box A are always
true. It does not matter whether the factors
and the product are whole numbers, fractions,
or mixed numbers.
4. Finish the examples in box B.
Find n in Ex. 5-12. Show all your work.
5. 3 X i = n; n =
6. n = \ X 6; n =
7. 5 = n — 3; n =
8. 10 — n = 2; n =
102
9. X n = 10; n =
10. n X li = 4J; n =
11. i = n - 3; n =
12. | X n = 15; n =
Solving Problems
Sometimes, when you must solve a prob-
lem, it is hard to decide whether you are to
find a missing product or a missing factor.
Here is a way that may help you:
a. Let n stand for what you must find.
b. Use n and the numbers given in the
problem to write in arithmetic language exactly
what the problem tells you.
c. Think what to do to find n.
d. Find n.
This method of statements with n is used in
the chart below to help you solve problems 1-7.
Complete the solutions.
1. Mr. Ames drove 240 mi. in 6 hr. How
many miles did he average per hour?
2. Edna used 2 yd. of ribbon from a piece.
This was \ of the ribbon in the piece. How
much ribbon had there been?
3. After the Maxwells had driven 150 mi..
Dad said, “Well, we’ve gone f of the way.”
How far were they going on the trip?
4. How many feet are there in 17 yd.?
5. Sue took 12 of her 42 snapshots at the
beach. What part did she take at the beach?
6. When Esther divided her books into 4 piles,
there were 3 books in each pile. How many
books did she have in all?
7. Mother said, “You boys may have \ of
the cookies in the jar.” If the boys took 16,
how many cookies had been in the jar?
, jj ■ — — — — — -
Let n stand for
The problem says
To find n
” — ■ — - — , 1
n = ?
1. Average number of
miles per hour.
1
6 X n = 240
Divide
n = 240 -T- 6 =
II
2. Number of yards in
the piece.
n = 2
n =2 4- 1 =
3. Number of miles in
the trip.
| X n = 150
n =
4. Number of feet in
17 yd.
n =
5. Part taken at the
beach.
n X 42 = 12
Divide
n =
6. Number of books in
all.
n =
7. Number of cookies
in the jar.
n =
703
Finding II, the Missing Number
Suppose you have this problem.
After Ed spent 30^ he had 45 £ left. How
much money did he have at first?
If you let n stand for the money he had at
first, you can write:
n — 30^ = 45^
Then you can find n by adding.
[A., S., M., D.]
Or, of course, you might state the problem
this way:
n = 30^ + 45^,
which makes it even easier to find n.
If you can easily find n, the missing num-
ber, in statements like these, you can solve
many problems by the method on page 103.
Find n, the missing number.
Show all the steps you take.
a
b
c
d
1. n + 6 = 14
5 = n + 2
4 =n X|
n -4
= 9
n =
n =
n =
n =
2. t + n = i
_JL_ n — i
10 11 — 2
| =3 -n
n +f
- 1^
n =
n =
n =
n =
3. 7 - n = 2
12 = 16 - n
i = n -r- 6
n v4
= 3
n =
n =
n =
n =
4. 10 - n = 5
4 = 8 A-n
9 'In = J
4 = n
-2
n =
n =
n =
n =
5. 2\ = n + \
i X n = 5
8 + n = 10
10 = n
L +4
n =
n =
n =
n =
6. n — f = 1
nX5 =15
li = i + n
n + 3
-
n =
n =
n =
n =
7. 20 = 4 X n
i - n - 2\
n -i =8
nXf
= 6
n =
n =
n =
n =
104
Line Graphs
1. Graph A shows how Jim’s puppy increased in weight
during his first 6
2. The horizontal scale shows the ,
and the vertical scale shows the
3. The weight scale shows that 1 small space is equal to
oz.
4. To read the graph, follow the arrows.
0 1 2 3 4 5 6
Months
How Jim’s Puppy Grew
a. When the puppy was born, he weighed oz.
b. At 3 mo. the puppy’s weight was
c. The puppy gained more weight between mo. and
mo. than during any other month.
5. Complete table B below the graph.
6. Could you show the puppy’s weight by a bar graph?
Explain how.
Age
Weight
At birth
10 oz.
1 mo.
2 mo.
3 mo.
4 mo.
5 mo.
6 mo.
Graph C shows, to the nearest 5 feet, the number of feet a
car travels after the brakes are applied.
7. The numbers on the horizontal scale show speeds in
per
8. On the vertical scale, one of the small spaces stands for
ft.
9. The graph shows that at 20 mi. per hour a car travels
ft. before stopping. Complete the table below.
Speed in miles per hour
20
30
40
50
60
Feet car travels
0 10 20 30 40 50 60
Miles per hour
Stopping Distance
for Car after
Brakes Have Been Applied
105
Drawing
As you have found, it is easy to read a bar
graph or a line graph. Sometimes, however,
it takes a bit of figuring to decide what scale
to use in drawing a graph.
If you have small numbers to show on a
graph, you can let 1 small space on the num-
ber scale stand for 1 unit. If the numbers
are large, it is usually better to round them
first and to let 1 small space stand for more
than 1 unit.
To decide on the scale, look at the smallest
number and the largest number in the group
of numbers to be graphed.
In figuring the following, suppose that you
will use the graph paper below:
1. On this graph paper there are di-
visions to the inch.
2. If the smallest number to be shown on
a line graph is 2 ft. and the largest is 20 ft.,
you might let 1 small space = 1 ft. Then the
highest point on the graph would be how many
spaces high?
3. In Ex. 2, if you let 1 space = 2 ft., then
the point for 20 ft. would be spaces high.
A point representing 11 ft. would be
706
Line Graphs
4. If you want to graph numbers of which
the smallest is 3,208 ft. and the largest is
17,849 ft., you had better round the numbers
to the nearest
5. If you round 3,208 and 17,849 to thou-
sands, you get thousand and thou-
sand for your lowest and highest points.
6. Then if you let 1 small space = 1 thousand
feet, the highest point on your graph (for
17,849, rounded to thousand) will be
spaces high.
On the graph paper below, draw line graphs
to show the facts in these tables:
7. Pete sold these papers last week:
Mon.
Tues.
Wed.
Thurs.
Fri.
Sat.
20
30
25
35
50
40
8. The Charity Fund grew as follows:
At end of — *
1 wk.
2 wk.
3 wk.
4 wk.
Total — y
$1,619
$2,760
$4,842
$5,716
o
A Review of Fractions
Do what the signs tell you to do.
a b c d e
!• Te
24
00
4^ 1 •— *
4I
t'- 1 OO
<N
+
+ io|
+ 9f
+ 5l
+ 2|
2. 10|
24§
4
4
7\
1
CO
o 1 ^
-8 9 10I
-4
-4
— 2-
z4
a
b
c
3. f X 45 X| X|
=
ft x f X f X 20 :=
6 X j X |
Xf =
4. § + 6 =
14 + If -
3 ^ 11 _
7*a2 —
5. 24"* | =
8 +t =
^3 ^ 17 _
->4 . 18 —
For each problem tell whether you will add, subtract, multiply, or
divide. Then solve, and draw a ring around the answer.
6. Ava spent § hr. making apricot sauce. After soaking the Space for Work
dried apricots, she simmered them for ^hr. How long did
she soak them?
A.? S.? M.? D.?
7. The comic strips in a newspaper just fit across 4 columns
of newsprint. If these comic strips are 1\ in. wide, how wide
is a column of newsprint?
A.? S.? M.? D.?
8. A square skillet, 10J in. on a side, has a 4|-inch handle
on one side. How long is the skillet including the handle?
A.? S.? M.? D.?
9. A strong paste for mending books can be made by mixing
1 tsp. flour, 2 tsp. cornstarch, and \ tsp. powdered alum, add-
ing 3 oz. water, and cooking in a double boiler until thick.
[Find a. the ratio of flour to cornstarch; b. the ratio of alum
to cornstarch.
A.? S.? M.? D.?
707
wi>— kni w iw
Testing What You Have Learned
Add, subtract,
multiply, or divide,
as the signs tell you.
[Cumulative Review]
a
b
c
d
e
1. $23.06
+ 15.98
5 ft. 8 in.
+ 2 ft. 6 in.
3 lb. 9 oz.
+ 4 lb. 8 oz.
4,9 12
+ 2 6,397
395,428
+ 872,906
2. $85.98
-27.50
70,6 52
- 9,084
$130.75
-22.15
6 lb. 3 oz.
— 5 lb. 9 oz.
8 ft. 9 in.
— 4 ft. 10 in.
3. 3,2 16
X 4 5
$6.75
X 3 2
4 ft. 3 in.
X 5
3 qt. 1 pt.
X 2
X 2
Write any remainder in a fraction in best form.
4. 72)4 ,116
48)1 ,830
4)5 yd. 1 ft.
2)3 hr. 2 0 min.
6)$ 1 0 . 4 4
5. Round as directed: a. 1,784 b. 32,465
To the nearest thousand
To the nearest hundred
To the nearest ten
6. From 6f, f, and 9, choose an example of a
a. Mixed number e. Denominator
b. Numerator f. Dividend
c. Divisor g. Fractional unit
d. Proper fraction h. Improper fraction
Give the smallest common de-
nominator for
7. J’s and J’s.
8. i’s and ^’s.
Find the ratio of
9. 3 to 9.
10. 8 to 2.
11. 4 ft. to 5 ft.
12. 6 in. to 1 ft.
Write in figures the numbers in Ex. 13-15.
13. Six hundred fifty- two thousand four hundred thirty-five
708
14. Two billion nine hundred sixty thousand four hundred fifty
15. Three million six hundred thousand
16. Round to the nearest whole number,
a. 15J b. 33J c.4*_.
17. Change
a. § to sixths.
b. 3f to an improper fraction.
c. to best form.
d. xf to lowest terms.
18. In a picture of a dog to the scale
the dog’s tail is drawn J inch long. How long
is the real dog’s tail?
19. Joe weighs 120 lb. His father weighs
160 lb.
a. Joe’s weight is what fractional part of
his father’s?
b. Joe’s father is how many times as heavy
as Joe?
Divide. Use cancellation when you can.
20. If - 6 =
21. 3J - 5 =
22. 4f -r- 7 =
23. 8§ - 4 =
24. 9f - 8 =
Divide. Give answers in best form.
9 - 5 5 _
6 • 9 —
28. * -5- 1 =
29. 4J - 1* =
30. 2f T If =
Find answers for problems 31-34.
31. Sara paid 16^ to mail a parcel. The
postage rate was 10^ for the first pound and
1J^ for each additional pound. How much
did the parcel weigh?
32. Doris put 9 books, each Jin. thick, to-
gether on the library shelf. How much shelf
space did they take?
33. Sue said, “My father is 2f times as old
as I am.” Sue is 12 yr. old. How old is Sue’s
father?
34. Ralph has used 13 pages of his album to
mount 234 stamps. Find the average number of
stamps to a page.
109
New Marathon
Record Weather p '“T
HIGH MILEAGE SHOWN j-^Ulk P
NEW YOR&^April 2
tough course^6.8jni. lo/breaJc
Enright brok&-atr recor, _
in this annual marathon evcm..
The course is longer than the
hn
■Weat'1-7o%"frer,Sdar
B
3.6 jmi.
Roller Skating Races iCes Up
NEWTOWN, MO., April 21— hnPril 2l
P KpoL f > T • f*UV~
The best time in this ye^-«-*miior
Championship races wafi56 5)sec
the record of Ted Abram ^
i . iAwtuiuo, ex incur
|ber of the Pierce School in this
Tnem- ln Price
Understanding
Miss Bryan’s pupils looked for decimals in
the newspaper. They found the items above.
1. All the decimal fractions in these clip-
pings are They have
the same fractional unit,
2. Write the ringed numbers in these clip-
pings with common fractions.
a. 26.8 = b. 2.7 =
Decimal Fractions
c. 3.6 p d. 56.5 =
e. 0.7 =
3. In 0.7, the 0 means
4. The mixed decimal 3.6 is the sum of
the whole number ___ and the decimal fraction
Written with a common fraction, 3.6 is
, or in best form
• 1.2
I
I i i "i i ] i i r i y i r i i t "i r
0 0.5 1.0 1.5
5. Mark with an arrow and
number line above:
a. 1.6 b. 0.3 c. fo d. 1.9
6. Complete the table below to show count-
ing by tenths from 3.0 to 4.0
3.0, or 3
3.1, or 3yo
3.2, or 3j%, or 3^
i ~i — | — | — i — | — j — | — i — i i i — [■ — i — i — r — r — | —
2.0 2.5 3.0 3.5
label the following points on the
e. 2.4 f. 3.1 g. 2* h. 3.4
Match the decimals below to Ex. 7-16.
1.1 0.8 101.2 8.0 4.2
20.4 2.4 0.2 100.1 10.1
7. One and one tenth
8. Four and two tenths
9. Eight tenths
10. Twenty and four tenths
11. Ten and one tenth
12. Eight
13. One hundred and one tenth
14. Twenty-four tenths
15. One hundred one and two tenths
4.0, or
770
16. Two tenths
■gjjM
|lip*ar Mflfc
w. i W ^
High Tide, Low Tide, and Decimals
1. One day in April at San Diego, Cali-
fornia, the height of high tide was 3.8 ft.
The height of high tide „
at Boston, Massachusetts, A B 1
was 5.9 ft. more than this, 3^ 3.8
or ?_ ft. + 5tq + 3.9
Boxes A and B show — 9.7
two ways of adding. In 9^
adding decimals, carry 1
just as you do in adding whole numbers.
The height of high tide at Boston that day
was ft. Is the work in box A a check
for the work in box B?
[A. and S. of tenths ]
2. In a drawbridge across a small tidal
river, the draw rises 36.2 ft. above aver-
age low tide and
27.6 ft. above aver-
age high tide. Find
the difference be-
tween high and low
tides.
Boxes C and D show the work. In box D,
do we borrow just as we do in subtracting
whole numbers? The differ-
ence is ft.
— — —
A
B , !
J10
3.8
+ 5yo
+ 5.9
017
°10 ~
9.7
| 9*
c
36* = 35f§
-27* =27*
D
m
-27.6
8*
8.6
Add or subtract. Try to think the carrying and borrowing.
a
b
c
d
e
3. 6 . 4
I +6.3
4.2
+ 1.7
5.5
+ 0.9
2.3
+ 10.0
97.3
+ 3.4
4. 8.5
-6.2
9.0
-3.5
2.6
-0.7
69.1
-55.8
43.7
-36.4
5. 6.8
+ 2.5
1 . 6
-0.8
6.3
-3.7
18.7
+ 35.4
55.2
-35.3
6. 4.9
+ 8.3
8.4
-7.8
7.7
+ 8.9
97.6
+ 9.9
60.1
-37.6
f
12.6
+ 5.1
g
6.6
+ 27.7
h
3 76.8
+ 276.4
i
162.9
+ 653.7
77.4
-36.7
62.9
-45.5
653.8
- 325.3
338.4
- 242.9
89.3
+ 64.6
82.0
-63.6
102.6
-64.2
2 18.3
+823.8
68.0
+ 5 1.9
3 1.8
-22.7
963.3
- 899.8
270.5
+ 904.0
111
Tenths, Hundredths, Thousandths
B C
[ Meaning ]
1. Square A is divided into equal
parts. So 1 of the equal parts is , or 0.1,
of the whole square.
2. Color square A: a. yq red; b. 0.1 black.
3. What decimal fraction of square A is not
colored?
4. Are the equal parts of square B larger
or smaller than those of square A?
J
1
J
~ r
J
P
1
1
U
r
q
n
...
j
□
' (
[
P
r
P
u
'
~p
T
T
T
UT
ltt
J
m
±
T
It
h
n
FT
|
LL
IT
10. Each part of square C is —
square. The decimal form is 0.001
11. Write j;
of the
Yoo as a decimal fraction.
12. Color 0.006 of square C red.
13. Complete the following table:
5. In square B there are equal rows,
with equal parts in each row, making
equal parts into which the whole square
is divided.
6. Each part of square B is — - — , or 0.01,
of the whole square. We read 0.01 as “one
hundredth.”
7. 0.05 = -jqq. Color 0.05 of square B red.
8. 0.10 = — . Make 0.10 of square B
black.
9. Square C has rows of equal
equal parts in all.
Common
Decimal
Fraction
Fraction
7
1 0
}
309
1,000
3
100
77
1,000
50
1 00
:
0.39
0.9
0.057
0.003
0.30
parts each, or
112
Adding and Subtracting Hundredths and Thousandths
[Like-fractions]
A
0.84 1.44
+ 0.08 +0.09
B
0.84 1.44
-0.76 -1.06
C
1.375 6.250
+ 1.375 - 2.750
1. Ed Myers collects United States com-
memorative stamps such as those shown above.
The catalogue issued by the United States
Post Office Department lists the dimensions
of the printed part of these stamps as 0.84 in.
(high) by 1.44 in. (wide).
Ed estimated that the perforations increase
the height of a stamp by 0.08 in. and the width
by 0.09 in.
Find the height and width including per-
forations. Finish the work in box A.
2. The Pan-American stamps of 1901 are
0.76 in. high and 1.06 in. wide, not including
perforations. Find the difference in the dimen-
sions of these stamps and the stamps shown
above. Finish the work in box B.
3. Ed wanted to make a special sheet of a
few stamps as a present. Each stamp was
1.375 in. wide. If he mounted 2 stamps on a
sheet 6.250 in. wide, how much space would be
left? Finish the work in box C.
Add in rows 4 and 5.
a
b
c
d
e
f
g
4. $1.65
9.78
5.84
$4.63
$7.50
6.333
3.875
+ 0.32
+ 1.14
+ 4.16
+ 0.87
+ 4.10
+ 1.420
+ 5.125
5. 4.76
7.24
9.17
$5.05
$6.25
9.378
7.364
+ 7.92
+ 3.99
+ 4.65
+ 0.98
+ 7.89
+ 2.375
+ 7.45 1
Subtract
in rows 6 and 7.
6. $3.75
$9.60
3.98
$1.75 $0.75
0.740
1.351
-2.40
-3.75
-3.25
-1.55 -0.33
-0.452
-0.366
7. $9.36
3.14
5.00
$4.0 1
2.15
5.652
7.733
-6.76
- 2.42
-3.64
-0.73
-0.69
-4.548
-3.849
773
Adding and Subtracting When
Complete the following table to show equiv-
alent (equal) fractions:
Tenths Hundredths Thousandths
1. 0.1 = 0.10 = 0.100
2. 0.3 = 0.30
3. 0.7 =
4 = 0.40 =
5 = = 0.800
6 = 0.90 =
7 = = 0.200
8. 0.5 = =
Decimal Fractions Are Unlike
[ Exact decimals ]
Write each number in this list beside the
number in Ex. 9-20 which it equals.
0.01 3.5 2.02 0.2
2.0 0.1 0.35 1.0
35 2.75 0.02 202
9. 2.020 15. 3.50
10. 35.0 16. 0.20
11. 1.00 _ 17. 202.0
12. 2.750 18. 0.10
13. 0.010 19. 0.350
14. 2 20. 0.020
A Common Fractions
Decimals
B Common Fractions
Decimals
3
300
0.3 = 0.300
5 5 0 0
0.5 = 0.500
1 0
— 1,000
10 “ 1,000
1
1 0
0.01 - 0.010
75 7 5
- 0.075 = 0.075
100
— 1,000
1,000 — 1,000
I 7
7
+ 0.007 - 0.007
425
0.425
1 1,000
~ 1,000
1,000
317
1,000
0.317
Fractions to be added or subtracted must be like-fractions.
First study the additions and subtractions in boxes A and B.
Then in each example below, write zeros to give all the numbers the
same fractional unit.
Then add or
subtract as directed.
a
b
c
d
e
f
21. 0.6
0.39
+ 0.02
0.825
0.04
+ 0.5
0.95
2.6
+ 1.375
4.04
3 . 3
+ 2.767
11.33
6.063
+ 4.9
8.8
0.08
+ 9.9
22. 0.53
-0.2
6.7
-4.33
9.75
-3.936
3 .
-1.45
8.063
-3.8
7.4
-4.267
23. 9.8
-4.27
3.81
+ 6.5
7.424
+ 1.81
4.753
-2.9
5.49
-5.049
9.186
+ 2.9
24. 0.4
+ 9.72
7 . 3
-6.444
4.
-0.667
8.82
-3.142
56.71
+ 8.8
3.04
-1.206
774
Comparing
Here are some quick ways to tell whether
one decimal is larger or smaller than another:
1. You know that 0.7 is more than 0.5 be-
cause 7 are more than
5
2. 0.7 is more than 0.64 because 0.7 equals
hundredths, and hundredths are
jmore than hundredths.
3. 0.361 is more than 0.36 because
thousandths are more than thousandths.
4. Any mixed decimal, like 1.1, is more
than any decimal fraction, like 0.879, because
a decimal fraction is always less than
5. If two mixed decimals have different
whole numbers, the one with the
whole number is the larger number.
4.795 is than 5.1
Draw a ring around the larger number in
iach of these pairs of numbers:
6.
0.75
0.85
7.
0.23
0.023
8.
0.995
9.95
9.
0.87
0.89
10.
0.04
0.4
11.
50
5.00
12.
1.98
2.1
13.
0.6
0.61
Decimals
In each of rows 14-19, underline the number
that is smallest.
14. 0.45
0.045
4.5
15. 1.2
0.12
0.012
16. 59.035
59.3
59.35
17. 0.879
2.98
3
18. 40.4
4.04
0.404
19. 22.66
26.26
26.62
Write a number that is
20. Smaller than either 0.08 or 0.6
21. Larger than either 6.5 or 7.29
22. More than 0.05 and less than 0.5
23. Less than 2.4 and more than 2.3
24. More than 0.5 and less than 1
25. Larger than 2.8 but smaller than 3
26. Smaller than 5.15 but larger than 3.09
27. Less than 24.0 and more than 14.9
28. Smaller than 30.2 and larger than 25
29. More than 7.008 but less than 7.1
30. Between 40.4 and 40.45
775
Find the Mistakes!
For a quick test, Miss Bryan wrote some
examples on slips of paper and let each pupil
work one. Check each pupil’s work. If it is
not correct, copy the example and work it
correctly. Write the correct answer on the
line after each example.
6. Laurie: 0.61 + 5 + 7.75 =
0.61
5
+ 7.75
8.41
7. Carol: 3.48 - 1.3 =
3.48
-1.33
2.15
1. Bill: 1.375 +2.5 =
1.375
+ 2.005
3.380
8. Dan: 0.2 + 0.07 + 0.03 =
0.02
0.07
+ 0.03
0.12
2. Pete: 1 + 0.01 =
9. Sam: 8 — 2.5 =
1 .00
+ 0.0 1
1.01
6 5
3. Bob: 4.6 - 2.80
10. Lorna: 10.4 + 1.04 =
4.60
-2.80
1.80
1 .04
+ 1.04
2.08
4. Joe: 6+1.8 =
11. Sally: 40 - 0.44 =
0.06
+ 0.18
0.24
40.00
-0.44
39.56
5. Anne: 0.38 + 2.26 =
0.38
+ 2.26
2.64
12. Ben: 85 - 8.5 =
85
- 8.5
0
116
Rounding
^ 0 0.1 0.2 0.3 0.4
| i i i i I i i m | m i i | m i i | i i i i i i i ,rr|~r~r~n "["r i tt| i m i | i i i
Bo 0.10 0.20 1 0.30 0.40 '
0.28
1. Suppose you want to round a mixed deci-
mal, like 5.8, to the nearest whole number.
You round mixed decimals in the same way
that you have rounded other numbers.
For 5.8 you must decide whether the decimal
fraction 0.8 is nearer to 0 or to
2. Find 0.8 on scale A above. You see
that 0.8 is nearer to 1 than to So, to
the nearest whole number, 0.8 is rounded to
i
3. Rounded to the nearest whole number,
5.8 is
4. Use scale A to help you round these
decimal fractions to either 0 or 1. Remem-
ber that you round 0.5 upward.
b. 0.1 f. 0.4
c. 0.5 g. 0.2
d. 0.7 h. 0.6
5. The fractional unit on scale B is
In 1 there are hundredths;
that is> 1 = ToT-
6. In 0.1 there are hundredths.
7. 0.28 lies between 0.2 and ; it is
nearer to than to ; so rounded
to the nearest tenth, 0.28 is
Decimals
[Meaning]
0.5 0.6 0.7 0.8 0.9 1
j | m i i | i i i i | i i 1 1 | i M i | i i i i | i i i i J n i i | i i i i 1 1 i i i | i i i 1 1
0.50 0.60 0.70 0.80 0.90 1
8. Rounded to the nearest tenth, 0.23 is
9. Show these points on scale B, and then
round each number to the nearest tenth:
a. 0.17 e. 0.07
b. 0.61 f. 0.36
c. 0.74 g. 0.52
d. 0.99 h. 0.85
10. Round each of the following decimals to
the nearest tenth:
a. 16.34 e. 2.66
b. 5.05 f. 19.75
c. 4.41 g. 6.70
d. 8.98 h. 10.83
11. Round to the nearest whole number:
a. 13.5 e. 18.2
b. 8.62 f. 9.9
c. 3.4 g. 7.35
d. 1.58 h. 0.73
12. Using what you have learned about
tenths and hundredths, round each of these
decimals to the nearest hundredth:
a. 0.321 f. 5.398
b. 0.148 g. 0.515
c. 6.476 h. 2.902
d. 0.094 i. 8.009
e. 9.207 j- 4.197
777
Using Measurement Numbers
One day Bob helped his father check the
figures on a land survey. His father asked Bob
to add 814.37 ft. and 732.5 ft.
814.37 ft.
814.4
+ 732.5 ft.
+ 732.5
1,546.9 (ft.)
Bob thought he should write 732.5 as hun-
dredths before adding, but his father told him
that instead he should round 814.37 to tenths
because these are measurement numbers.
1. Bob’s father said that 732.5 ft. is a less
accurate measure than 814.37 ft. because the
fraction 0.5 might have been rounded from a
fraction as small as 0.45 or as large as
[A. and $.]
2. The hundredths that can be rounded to
0.5 are:
, , , ,
, , , , .
3. So, unless you know that the distance
732.5 ft. was actually 732.50 ft. to the nearest
hundredth, you should not change the decimal to
hundredths. To make the measurement num-
bers 814.37 ft. and 732.5 ft. have the same
fractional unit, you round 814.37 to
4. Of the measurement numbers 814.37 and
732.5, the one with the larger fractional unit is
Before adding or subtracting measurement decimals, find the one
with the largest fractional unit. Express all the other numbers in the
group with this fractional unit.
Add or subtract these measurement numbers:
a
b
c
5. 3.2 in.
8.312 in.
5 0.28 mi.
+ 5.75 in.
+ 3.25 in.
+ 8.9 mi.
6. 8.6 oz.
— 2.86 oz.
6.25 in.
-4.125 in.
9.833 yd.
— 4.50 yd.
7. 1 2 . 3 yd.
+ 2.67 yd.
1 . 1 1 4 ft.
+ 0.75 ft.
0.625 mi.
+ 1.4 mi.
778
Decimals and Their Equivalent Common Fractions
Sometimes it is easier to use a common
fraction like \ than a decimal fraction like
0.25. So it is convenient to be able to change
a decimal fraction to a common fraction.
1. Look at box A. 0.25 means
hundredths, which, as a common fraction, is
25 . .
vritten , or, in lowest terms, .
2. In box B, how was yo changed to
3. To change -^ioo t0 lowest terms (box C),
rou could first divide both 375 and
)y 5; then you could reduce the resulting
Finish the work started below.
375 = 75
1,000 ~
You can change t0 lowest terms in
one step. You may have noticed that 375 is
exactly 3 X 125. See if 125 is contained in
1,000 without a remainder. 1,000 -=-,125 =
Then -3-7-5 —
i nen 1,000 —
4. $0.50 is what fractional part of a dollar?
Show all the steps to prove it.
0 — 25 — i
— lOO — 4
B
OS § 1
U.3 - 10 - 2
D
0 S7S — 375 -
U.J/3 — lj0oo ~
0 10— *° — -1-
v/.iv/ — 100 — 10
fraction again.
. 0.90 =
I
. 0.75 =
Change the following to common fractions in lowest terms. Show
all the steps you take.
9. 0.60 =
10. 0.025 =
11. 0.15 =
12. 0.2 =
13. 0.625 =
14. 0.80 =
15. 0.48 =
16. 0.232 =
17. Finish the tables below.
Halves and Fourths
0 ?S — 25 — i.
U-ZJ — 100 4
0.50 =
0.75 =
100
75
Fifths
0.20 = = -±-
0.40 =
Eighths
01 ?S — 125 — 1
— 1,000 — 8
0.375 =
0.60 = _
0.625 =
0.80 =
0.875 =
I
779
Multiplying Decimals
A
B
c
0.7
7 tenths
0.7
0.7
X 3
X 3
+ 0.7
21 tenths,
' 2.1
2.1
or 2.1
1. Sandra had 3 candy bars. Each bar
weighed 0.7 oz. All 3 bars weighed how many
ounces? Boxes A-C show three ways to find
the answer. Write it below.
by Whole Numbers
[Tenths]
2. One of the small cakes of soap used in
hotels weighs 2.9 oz. How much do 3 cakes
weigh? Boxes D and E show two ways to mul-
tiply. Write the answer below.
Study the work in boxes C and E until you understand how to
multiply a decimal fraction or a mixed decimal by a whole number.
Then do the work in rows 3-5.
abed
3. 5 X 0.1 = 4X0.9= 5X0.6= 5x0.8 =
4. 7 X 0.2 = 3 X 0.4 = 4 X 0.2 = 6 X 0.3 = _
5. 3 X 0.5 = 9 X 0.1 = 6 X 0.7 = 7 X 0.4 =
When a number with the fractional unit ^ is multiplied by a
whole number, the product must show tenths.
a
b
c
d
e
f
g
h
i
6.
1 . 3
7.2
2 . 2
3.5
5 . 7
8.4
7 . 5
4.5
6.4
X 4
X 5
X 9
X 3
X 6
X 2
X 3
X 2
X 7
7.
4 . 7
5.4
3.6
7.4
2.7
1 .5
8 . 1
6.6
2.5
X 3
X 9
X 4
X 2
X 4
X 5
X 5
X 2
X 3
In each of Ex. 8-
-17, draw a ring around the correct answer.
8.
3 X 1.2
= 3.6
0.36
36
13.
5 X 4.1 =
2.05
20.5
0.205
9.
2 X 3.5
= 70
0.7
7.0
14.
2X6.0 =
12.0
0.12
1.20
10.
8 X0.1
= 0.08
0.8
8.0
15.
1 X 1.6 =
0.16
1.6
1.60
11.
3 X 0.8
= 0.24
2.4
0.024
16.
2 X 8.2 =
1.64
0.164
16.4
12.
5 X 0.5
= 2.5
0.25
25.0
17.
3 X 2.3 =
6.9
0.69
69
720
Multiplying Hundredths and Thousandths
[ Multiplier a whole number ]
A
$1.35
X 5
$6.75
C
$0.75
X 12
1 50
75
D
4.105
X 24
16 420
82 10
E
0.035
X 15
175
35
F
When a decimal is multiplied by a
whole number, the product has the same
fractional unit as the decimal factor,
and so must show the same number of
decimal places.
1
•
B
$2.98
X 20
$59.60
$9.00,
or $9
98.520,
or 98.52
0.525
Study the work in boxes A-E and the statement in box F. Then
work Ex. 1-30. Make sure that each product has the correct fractional
unit (the same number of decimal places as the decimal factor). Then
reduce the decimal fraction if possible.
1.
0.38
2. $6.04
3.
0.106
4.
0.007
5.
$0.75
X 3
X 3
X 9
X 8
X 7
6.
3.024
7. 0.40 7
8.
1 .4
9.
9 . 1
10.
0.018
X 6
X 4
X 5
X 6
X 7
11.
0.19
12. 12.7
13.
0.06
14.
$3.49
15.
$8.05
X 1 2
X 2 6
X 9 2
X 9 0
X 9 6
16.
2.375
17. $8.75
18.
1 .048
19.
0.265
20.
4.5
X 3 8
X 8 0
X 1 8
X 1 6
X 2 1
21.
$5.32
22. 0.45
23.
0.09 1
24.
$1.54
25.
0.345
X 5
X 4
X 9
X 2
X 9
26.
12.7
27. 0.375
28.
$1.98
29.
0.019
30.
$3.45
X 4 9
X 4 2
X 2 7
X 3 5
X 7 0
727
Multiplying by 10 or 100 or 1,000
[ Multiplying a decimal ]
B
C
6.
2.7 5
0.0 3 4
a. 10 X 6.
= 6 0.
a. 10 X 2.75
= 2 7.5
a. 10 X 0.034 = 0.3 4
b. 100 X 6.
= 600.
b. 100 X 2.75
= 2 7 5.
b. 100 X 0.034 = 3.4
c. 1,000 X 6.
= 6,0 0 0.
c. 1,000 X 2.75
= 2,7 5 0.
c. 1,000 X 0.034 = 3 4.
Boxes A-C show an easy way to multiply by 10 or 100 or 1,000.
You look at the number to be multiplied and think of the decimal
point as moved to the right as many places as there are 0’s in the
multiplier.
Work this way in writing products for rows 1-5.
a b c
1. 10 X 6.1 = 100 X 4.75 = 1,000 X 2.875 =
2. 100 X 10 = 1,000 X 6.4 = 10 X 0.08
3. 1,000 X 0.06 = 10 X 46 = 100 X 0.3 = ___
4. 100 X 0.035 = 1,000 X 37.2 = 10 X 3.25 = ___
5. 10 X 0.5 = 100 X 8.5 = 1,000 X 0.9 = __
Meaning of Decimals
1. Is 0.03 larger or smaller than 0.3?
2. Does 4.2 equal 0.42?
3. Does 0.85 equal 0.850?
4. Is it true that 1.75 = If?
Write in figures as decimals:
5. Two hundred thousandths ___
6. Five tenths
7. Seventy-five hundredths
8. Two and nine tenths
9. Six thousandths
10. Eight hundredths
11. Twenty and four thousandths
722
In each row, draw a circle around the small-
est number and a square around the largest
number.
12.
0.125
1.2
11.5
13.
0.89
0.089
8.09
14.
2.47
27.4
0.742
15.
5.2
0.520
5.02
Copy each of the numbers in rows 12-15 in
the column below in which it belongs.
Tenths Hundredths Thousandths
Multiplying by a Decimal
[The multiplicand a whole number ]
1. Boxes A, B, C, and D show
that if one of two factors is a
whole number and the other is
a decimal, the product has the
same fractional unit as the
2. Boxes C and show
A
35
B
249
C
85
D
0.375
X 2.4
X 0.13
X 0.375
X 85
14 0
7 47
425
1 875
70
24 9
5 95
30 00
84.0,
32.37
25 5
31.875
or 84
31.875
that if you change the order
Place decimal points correctly in the products below.
of the factors the product is
3. 1.05 X 267 = 2 803 5
5. 0.8 X 356 = 2 8 4 8
4. 0.125 X 28 = 3 5 00
6. 2.875 X 23 = 66 1 2 5
In the multiplications below, reverse the factors if it will make
the multiplication shorter.
a b c
7. 0.75 X 360 1.6 X 5 3.005 X 100
0.375 X 24
8. 3.04 X 25 0.86 X 182 2.125 X 16 4.2 X 154
9. 8.14 X 53 20.3 X 205 131.6 X 38 25.75 X 2
10. 0.046 X 121 6.4 X 750 1.25 X 125 98.5 X 10
723
Multiplying Decimals and Mixed Decimals
1 place 1 place
a. 0.2 X 0.6 =
b. 1.4 X 2.1 =
2 places
0.12
2.94
1 place 2 places 3 places
c. 0.3 X 0.25 = 0.075
d. 0.5 X 1.08 = 0.540, or 0.54
1. Use common fractions to check the ex-
amples in the box. It is easier, in this check,
if you do not cancel. The check for Ex. a
shows why.
a 2 v 6 12_ — 0 1?
a* 10 ^ 10 — 100 — V.l^,
b.
c.
d.
When two decimals are multiplied, the product has as many
places at the right of one’s place as there are decimal places
in both factors together.
Find the products. In rows 2-6, multiply without copying the
examples.
a
2. 0.4 X 0.8 = 0.2
3. 0.5 X 0.6 = 0.3
b
X 0.5 = 0.9 X 0.3 = ___
X 0.1 = 0.7 X 0.6 = ___
4. 0.4 X 1.1 = 0.2 X 3.3 = 1.3 X 0.2 = _
5. 2.2 X 0.9 = 0.9 X 0.01 = 0.06 X 0.6 =
6. 0.07 X 0.6
8.1 X 0.02 =
0.9 X 1.04 =
a
7. 9 7.4
X 0 . 0 4
b
2,98 7.5
X 0 . 5
c d
569.4 187.5
X 0 . 0 7 X 0 . 9
e
304.6
X 0 . 0 8
8. $6.95
X 2 4
123.7
X 0 . 3 5
42.08
X 1 .3
20 1.2
X 4 . 6
190.7
X 5 0 . 7
9. 14.9
X0. 1 8
46.1 28.4 59.3 3.72
X 0 . 5 9 X 0 . 6 1 X 8 . 4 X7.3
7 24
Decimals Smaller than Thousandths
a. One million three hundred twenty-six
thousand five hundred forty-seven AND eighty-
four thousand nine hundred seventy-five mil-
I lionths.
[ Meaning\
1. In chart A, write numbers b-i in figures
with each figure in its correct place.
b. One hundred fifty-six and fourteen hun-
dredths.
c. Twenty-four and forty ten-thousandths.
d. Twenty-five and thirty-five thousandths.
e. Fifteen hundred-thousandths.
f. One hundred six and five thousandths.
g. One hundred twenty-five thousand two
hundred fifty-two millionths.
h. Seventy-two thousand and nine tenths.
1. Five and nine hundred eleven millionths.
2. The name given to a decimal fraction is
the name of the last-used
B
124.5
2.020202
6.50
1.245
3.33333
3.6
0.1245
0.000002
12.45
5,140.5
4.3125
9.001
32.005
10.01010
0.125
3. From box B, copy each number into
one of the columns below under the heading
that tells the name of the decimal fraction.
Tenths Hundredths
Numbers in Which the Decimal Fraction Is
Ten-
Thousandths thousandths
Hundred-
thousandths
Millionths
4. Write the number 1.6875 in words.
“One and six eight seventy-five
5. Write 156.17667 in words.
125
Multiplying Decimals
[ Products with more than 3 decimal places ]
The examples below show the figures found by multiplying.
“Point off” these answers; that is, place each decimal point to show
the correct product. Sometimes you will need to write a zero or zeros
between the decimal point and the product figures in order to show
the correct number of decimal places.
1.
0.7 X 2.75 = 1 9 2 5
5. 0.12 X 0.0315 =
3780
2.
0.25 X 1.36 = 3 40 0
6. 0.125 X 9.82 =
122750
3.
0.375 X 0.625 = 2 3 4 3 7 5
7. 0.5 X 0.1875 =
93 7 5
4.
2.05 X 1.375 = 28 1 875
8. 0.036 X 0.9 =
324
When two decimals are multiplied, the product has as many
decimal places as there are in both factors together.
9.
Find products for rows 9 and 10. Be sure to place decimal points correctly.
a b c d e
2.43 1.75 3.106 4.012 1.0375
X 0 . 6 X 0 . 7 5 X 0.1 5 X0.1 25 X 0.2 5
10. 0.00 62 5
X 0 . 9
5.125 8.2
X 6 . 7 X 0 . 1 6
0.3125 1.1875
X 0 . 5 0 X 0 . 3
Review of Decimals
a
1. $3.64
+ 1.75
Add. When necessary, write zeros to make the addends like-decimals.
b c d e
0.375 4.3125 5.002 0.5
+ 3.333 + 0.625 + 6.02 + 1.98
[A . and $.]
f
9.12
+ 0.9
2. $0.75
-0.32
Subtract. First make sure that fractional units are alike.
2.6 12.5 8.01 7.3121
- 0.32 - 9.8 7 5 - 5.6 - 4.03
2.000
-1.667
126
A Test on Decimals
Change each decimal to a common fraction in lowest terms.
1. 0.25 = 3. 0.375 = 5. 0.01 =
2. 0.005 = 4. 0.0075 = 6. 0.125 =
Change to decimals.
Round to tenths.
11. 3.84 12. 1.16
Round to hundredths.
15. 0.206 16. 2.125
9.
875
1000
10.
106
1 0,000
13. 0.98
14. 0.07
17. 3.663
18. 0.005
19. Write the name of the place of each
figure in the number 2,148.3675.
1
Figure Place
2
1
! 4
; 8
3
6
7
5
[. -
20. For the number 4,075.1638 write
a. the figure in one’s place
b. the figure in tenth’s place
c. the figure in thousandth’s place
d. the figure in ten-thousandth’s place
e. the figure in hundred’s place
f. the figure in ten’s place
21. Write 16.00833 in words:
Write just the answers. Be careful about decimal points.
a
b
c
22. 0.2 + 0.1 = 0.9 + 0.2 = 0.001 X 2.5
23. 1.7 +0.6 = 1.5 -
24. 0.9 - 0.5 = 0.13
25. 1.6 - 0.7 = 1.4
26. 0.1 X0.01 = 0.02
-0.8 = 0.003 X 0.1 =
- 0.07 = 0.05 X 0.03 =
- 0.5 = 0.1 - 0.08 = _
X 0.4 = 10 X 4.75 = _
127
Vacation in Trailer Park
In boxes A and C, we used fractions to
show the division of the remainders. In
boxes B and D, we expressed the dividends
in decimal form and kept on dividing, so the
quotients show decimal fractions.
Find answers for the following problems.
Let your quotients show decimal fractions.
1. The Chapmans spent their summer va-
cation in their trailer. To reach Trailer Park,
they drove 279.3 mi. the first day and 287.7 mi.
the second day.
a. They drove mi. in all.
b. The daily average was mi.
2. They used 45 gal. of gasoline on the trip
(Ex. la). How many miles, on the average,
did they go on a gallon?
3. At Trailer Park, all trailers are 90 ft. from
the lake. Gary Chapman paced off this dis-
tance so that he could estimate the length of
his step. (The pace is the length of one step
from the toe of one foot to the heel of the
other.) Gary took 50 paces. What is the aver-
age length of his step?
4. Sandra Chapman took 36 paces in 90 ft.
How long is Sandra’s pace?
5. Sandra and Kendra rode on the merry-
go-round at the park. They bought a strip
of tickets at 10 rides for 75^.
a. How much does one ride cost on a strip
ticket?
b. They each took 2 rides. How much more
would their rides have cost if they had bought
single 10^ tickets?
6. Mr. Chapman paid $15.75 for 3 weeks’
rent of the space in the park. How much
was this by the week?
7. The Chapmans spent $17.50 for food one
week. What was the average daily expense for
food?
128
Changing Any Fraction to a Decimal
[ Whole number larger whole number ]
A
B
C
i = 1 -r- 4 = 0.25
|=3~8= 0.375
i = i
-i- 6 = 0.1 6§, or 0.17
0.25
0.375
0.1 6§
0.166
4lL00
sTTooo
6TTOOT
6JL000
8
24
6
6
20
60
40
40
20
56
36
36
40
4
40
40
36
4 2
Q ~ 3
4
Be sure that you understand how the common fractions in boxes A-C
are changed to decimals. Then change the following common fractions
to decimal fractions. Use as many 0’s in the dividend after the decimal
point as are needed to make the division come out even.
2 X d U =
8J5:
Change these fractions to 2-place decimals. Give each answer in
two ways — (1) as a 2-place decimal with the remainder in a common
fraction; and (2) rounded to hundredths.
or 6. Y2 = or 7. ye = or
729
Dividing a Decimal by a Whole Number
1. Sally bought three candy bars. Their
weights were f oz., 1.50 oz., and f oz. Find
the average weight of a bar.
Box . _ shows that the average weight is
oz., or about oz.
2. Box B shows that 0.12 3 =
3. In box C, explain why the dividend was
changed to 1.70
4. In box D, why were two 0’s used in the
dividend after the 7?
In the examples below, round the quotient to the nearest thou-
sandth if the division does not come out even in ten-thousandth’s
place or before that.
5. 5)2 7 . 4
6. 6)0 . 9 3
7. 9)3 6 2 . 2 5 8. 7)0 . 0 8 1 2
9. 12)0 . 0 7 10. 25)278 11. 34)0 .549
12. 48)0 .3125
130
Multiplying and Dividing by 10 or 100 or 1,000
A
B
10 X 4.75 = 47.5
47.5 -10 = 4.75
100 X 4.75 = 475.
475 - 100 = 4.75
1,000 X 4.75 = 4,750.
4,750 - 1,000 = 4.75
Notice that the multipli-
cations in box A are reversed
by the divisions in box B.
Study the boxes above. Then work Ex. 1-6
1. To multiply by 10, move the decimal point 1 place to the right.
2. To multiply by 100, move the decimal point places to the
3. To multiply by 1,000, move the decimal point places to the
14. To divide by 10, move the decimal point 1 place to the left.
5. To divide by 100, move the decimal point __ places to the
6. To divide by 1,000, move the decimal point
Write the quotients for Ex. 7-13.
Number
7. 250.8
8. 0.75
9. 22.67
0. 18
■
1. 2.125
2. 42.6
Divided
by 10
Divided
by 100
Divided
by 1,000
10 X 3.5 =
100X3.5 =
___. 1,000 X 3.5 =
3.5 - 10 =
3.5 - 100 =
places to the 3.5 — 1,000 =
Write the products for Ex. 14-20.
Number
Multiplied
by 10
Multiplied
by 100
Multiplied
by 1,000
14.
0.06
15.
2.1
16.
30.05
17.
7.624
18.
120.8
19.
0.1
20.
42.33
Dividing by a Money Number
1. For her party, Dianne spent $1.25 for
candy. If it cost $0.50 a pound, how many
pounds did she buy?
$0.50]$O5 = ^
100 x 1.25 125
100 X 0.50 ~ 50
50)125"
100
25
Change each example to one in which the divisor is a whole num-
ber. Write just the example; do not divide.
!■
$0.23)$1.75 = ) 4. $1.98)$10.95 = )
6. $2.30)$12 =
)
7. $0.48)$15.60 =
T
73?
Dividing by a Decimal Fraction
[Divisor:
tenths; hundredths; thousandths ]
A
0.3)6 = 3)60
20
3)60
6
0
B
0.04)03 = 4)30
7.5
4)303
28
20
20
c
0.012)33 = 12)3300
300
12)3300
36
00
D
0.08)0306 = 8)03
0.075
8)0.600
56
40
40
Study the work in boxes A-D. Then, in rows 1-4, write each example
again with the divisor changed to a whole number, and divide. If the
division does not come out even by thousandth’s place, round the
quotient to the nearest hundredth.
a
b
c
1. 0.6)9
0.09)1 . 8 9
k
00
0
0 1 2
2. 0.1)3 .24
0.04)2 3 . 6
0.3)7 .
26
3. 0.009)07 9
0.2)775
0.205)471
4. 0.78)677
0.875)1 . 2 5
0.24)1
.6
132
Dividing by a Mixed Decimal
34 +1.5=.?
22.666
15)340.000
30
40
30
10 0
90
1 00
90
100
90
10
Ans.: 22.67
B
8.019 # 1.03 | ?
7.785
103)801.900
721
80 9
72 1
8 80
8 24
560
515
45
Ans.: 7.79
75 ~ 3.25 = ?
0.176
325)57.500
32 5
25 00
22 75
2 250
1 950
300
Ans.: 0.18
In dividing by a mixed decimal, you work just as you do when the
divisor is a decimal fraction. First change the example so that the
divisor is a whole number. Study the work in boxes A-C until you
are sure that you understand it.
Divide in Ex. 1-6. Round quotients to the nearest hundredth if
they do not come out even by thousandth’s place.
1. 1.6)9.0 2
2. 1.134)3.9 2
3. 2.04)9 .12 5
4. 1.33)2.6 6
5. 1.01)7.37 5
6. 4.0)0 . 8 6
133
Decimals in Problems
Joe’s older brother told him about a useful
relationship in the circle.
To find the distance around a circle,
you can multiply the distance across it
through the center by 3.14.
b. How many more square inches are there
in one of the larger tissues than in one of the
smaller tissues?
5. A gram is a small unit of weight that
equals approximately 0.03527 oz. To the near-
est hundredth of an ounce, how many ounces
equal a gram?
6. The difference (called the “spread”) be-
tween the price per quart paid to the farmer
for milk and the price paid by the consumer
is 14.18 cents today. This spread was 9.55
cents eight years ago.
a. How much has the spread increased dur-
ing the eight years?
b. What has been the average yearly increase
in the spread? (Round your answer to the near-
est hundredth of a cent.)
1. Find the distance around a circle if the
distance across it through the center is 5 in.
2. Joe used the circle relationship to find
the distance around the stump of a tree that
was 2 ft. 9 in. across. Express 2 ft. 9 in. as a
mixed decimal and find the distance around the
stump, to the nearest 0.1 ft.
3. A meter is a unit of length that is about
39.37 in. A millimeter is 0.001 meter. To
the nearest 0.01 in., how many inches does a
millimeter equal?
4. A box of 9.00" by 9.75" cleansing tissues
costs 27^. A 25 box contains the same num-
ber of sheets 8.75" by 9.75".
a. Find the area of each size of tissue (to the
nearest 0.01 sq. in.).
7. Small fence pickets cost $0.06^ each if
bought at the lumber yard, and 7^ if deliv-
ered. Mr. Burke wanted 250 pickets. How
much would he save by buying the pickets
at the yard?
8. In his nature-study class, Toby learned
that the house sparrow weighs about 1.05
ounces, and the song sparrow weighs only
about 0.88 ounce.
a. The house sparrow is about oz.
heavier than the song sparrow.
b. The house sparrow is about
times as heavy as the song sparrow. (Round
your answer to the nearest tenth.)
c. The song sparrow weighs about 0 as
much as the house sparrow.
27^ size
134
; 25^ size
Three Kinds of Problems in M. and D
In working with problems, remember the
;hree kinds of problems in multiplication and
iivision. You learned about these with com-
In the boxes below, supply the missing words
and numbers.
Show your work in the column at the
non fractions.
right.
I. Finding the product.
0.07 X 15 = n
n = 0.07 X 15 = 1.05
You know the two factors.
To find the product, you
15
X 0.07
1.05
II. Finding the factor which shows a relationship, or
ratio; that is, how many times or what part of.
n X 15 = 1.05
n = : =
You know the product and
one factor.
To find the other factor,
you
III. Finding the other factor when the ratio factor is
given.
0.07 X n = 1.05
n = : =
You know the product and
one factor.
To find the other factor,
you
j
Find n. Show all your work.
a b
c
0.24 X 8 = n; n = n = 0.60 X0.5; n = n X 30 = 6; n =
n X 10 = 5; n = 0.50 X 9 = n; n = 4 X n = 2; n =
n = 0.15 X 3; n = 0.25 X n = 4; n = n X 60 = 15; n =
0.35 Xn = 7; n = n X0.6 = 1.8; n = 0.04 X n =3; n =
135
Review of Decimals
1. Margaret got the quotient $0.19633 when
she divided a money number. To the nearest
cent, how would she express this quotient?
Write in figures:
2. Nine thousandths __
3. Forty and six tenths
Change to decimals to the nearest thousandth.
6.
7.
8.
3
5 10 1000
Space for Work
Write as decimal parts of a dollar:
4. a. 38^ = $ b. 89^ = $
5. a. = $ b. 5^ = $ —
10. 10 X 3.6 =
11. 0.4 - 100 =
12. 0.03 X 1,000 =
13. 5.8 X 100 = __
Write only the answers for rows 10-13 below.
b c
1.9-100= 0.52 - 1,000 =
0.15 X 100 = $7.45 - 100 = .
$8.32 - 10 = 16 X 100 =
1,000 X $3.75 = 48 - 1,000 = _
Point off (that is, place the decimal point) to
show the correct product.
14. 0.07 X 95.2 =
15. 15 X 1.5 =
16. 2.75 X $100 =
17. 3.2 X 5.08 =
18. 0.2 X 0.375 =
6 6 6 4
2 2 5
$ 2 7 5 0 0
1 6 2 5 6
7 5 0
Round to the nearest whole number.
19. 3.3 2.49
20. 159.95 80.1 __
21. 17.2 65.89
22. 38.07 230.81
136
Change to common fractions in lowest
terms.
23. 0.3 =
24. 0.125 =
25. 0.48 =
26. 0.3125 =
27. 0.075 =
28. 0.01 =
29. 0.50 =
All Kinds of Numbers
Add, subtract, or multiply. Watch the signs!
a
b
c
d
e
1. 3 24
1,930
86
756
2.125
106
22 1
158
2,152
3.875
+ 55
+ 3,875
+ 750
+ 1,009
+ 0.50
2. 6 1
2i
<
+8!
+5I
+9I
+3I
3. 587
2,000
$16.52
13,406
6.805
- 398
- 956
-8.85
- 7,527
-3.96
7 -
8
236
1 .09
62.8
1,728
35.125
X 1 6
X 5 5
X 3 2
X 1 .5
X0. 18
3 w 2 -
4 A 3 —
7- 1 X | X f =
O 1 \/ 5 w 8
O. 2 A 6 A 15 —
>. 2\ X f X 8 = 10. i X 3^ X 7 X 7 =
Divide. In Ex. 11 and 12, round quotients to the nearest hundredth.
1. 23)7 . 8 9
12. 35)0 .597
IQ 2 _i_ 1 _
AO* 3*2
14. 3i + 2 =
737
Division with Decimals
At the right are examples that illustrate
all the kinds of work in division with decimals.
Just as with common fractions, some quotients
tell how many times and some tell what
part of.
Without dividing, write a check mark (y)
in the proper column to tell whether the quo-
tient will be a how-many-times number or a
what-part-of number.
Now work the examples in the boxes be-
low. When a quotient goes beyond thou-
sandth’s place, round it to the nearest
hundredth.
When you have finished dividing in each ex-
ample, look back at your check mark and see
whether the quotient is the kind of number you
thought it would be. If it is not, try to find and
correct your mistake.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Quotient will show
Example how many times what part of
1. 23 8
2. 3 : 4
3. 0.47 — 7
4. 12.6 — 5
5. 5.08 - 9
6. 34 - 0.8
7. 0.875 - 0.25
8. 0.36 - 0.75
9. 12.35-0.65
10. 17-2.125
11. 6 - 14.8
12. 0.5 - 3.2
13. 9.25 - 7.5
14. 4.02 -8.1
738
Right or Wrong?
Mark V in the box if a statement is correct, and X if it is
wrong. Read carefully, and think!
1. If you know a product and one of its two factors, you
multiply to find the other factor.
2. The name of a decimal fraction is the name of the last
decimal place used.
3. In a division example, any remainder is always smaller
than the divisor.
4. If you write two zeros after a whole number, you have
multiplied the number by 100.
5. If you know the number left and the number at first,
you subtract to find the number gone.
6. The Roman number that means 5 is X.
□
□
□
□
□
□
When a statement in the
left column is wrong, write
its exercise number on one
of the lines below and ex-
plain why it is wrong.
7. If you put a zero between the decimal point and the
tenth’s figure in a decimal fraction, you have divided the frac-
tion by 10.
8. Like-fractions must have the same numerator.
□
□
9. In dividing a decimal by a decimal, you first multiply
both dividend and divisor by a number that makes the dividend
a whole number.
; 10. The name of a decimal fraction that has three places is
“thousandths.”
11. To multiply a number by 10, you can think of the deci-
mal point as moved one place to the right.
12. To check a division example, you multiply the quotient
iand the divisor and add the remainder.
■
13. A whole number ending in two zeros is always 10 times
as large as a whole number ending in one zero.
14. Decimal fractions which have the same number of deci-
mal places are called “like-fractions.”
□
□
□
□
□
15. You point off the product in multiplying decimals by
giving the product as many decimal places as there are in the
multiplicand.
□
739
Short Division
[Even and uneven division ]
A
B
c
Long Way
Short Way
Long Way
Short Way
Long Way
Short Way
212
212
132, R3
132, R3
m
79J
4)848
4)848
7)927
7)927
6)476“
6)476“
8
7
42
4
22
56
4
21
54
8
17
2
8
14
3
1. In boxes A, B, and C, draw a big ring
around all work that you think but do not write
when you divide in the short way.
2. Write the quotient from box B with the
remainder in a fraction.
3. The remainder in box C is , so we
could write the answer as ,R
4. In box C, where does the J come from?
Use short division for the examples in rows 5-11.
a
b
C
d
e
f
g
5. 3)67
5)570
4)87
6)170
7)TT7
9)T89
5)375
6. 4)408
7JT47
8)876
9)579“
8)757
3)279
3)279
7. 5)10 5
7)779
6)T87
4)T77
5)547
6)677
4)T57
Write any remainder in a fraction in best form.
8. 4)74
6)79“
8)27
9)57
4)27
9)79
8)57
9. 9)67
5)76“
4)78
8)37
9)87
3)79
7)57
10. 4)77
8)77
6)27
7)57
6)47
9)60"
5)79
11. 5)2T
7)37
9)77
6)5T
5)37
8)4T
6)35
in
a
The following examples are the kind of subtraction you
. division examples. Write remainders.
bed
often find
e
12. 25 - 18 = _
____ 33
- 28 =
41
- 32 = ___
_ _ 63 - 56 =
71
- 66 =
13. 62 - 54 = _
52
- 45 =
34
- 27 = ___
20 - 12 =
54
- 48 =
14. 71 - 64 = _
____ 44
-36 =
22
- 18 =
__ 51 -49 =
31
- 24 =
140
D
68, R6
8)550
F
4.11 -4- 6 = ?
G
To the nearest 0.01,
2.15 4- 0.9 = ?
H
315 =75
I
f = 0.285 . . .,
E
457, R5
7)3)204
0.685
2.388 . . ., or 2.39
6)4110
9)21.500
or 0.29
15. Write the answers for Ex. D and E with
the remainders in fractions.
Ex. D Ex. E
16. The dots in Ex. G and I show that the 5)3 7 5
division does not come out even. The fifth
quotient figure in Ex. G would be
Divide, using short division. Write any remainder with R.
17. Work Ex. H and I in the long way and
draw a ring around the part that you do not
write when you work in the short way.
7]2~
a
b
c
d
e
f
18.
8)2 3 5
6]5~0~0
9)320'
7)5TT
8)70T
4)312
19.
7)601
9)5“02'
4)TT0
6)522
5)217
6)516
j
Rewrite examples if necessary. If any quotient is not
hundredth’s place, round to the nearest tenth.
even by
a
b
C
d
20.
6)2 , 0 5 8
0.4)1 . 0 1
0.6)5“
5)7 . 3 2
21.
0.8)3 0 . 5
7)3 , 2 6 5
9)4731 0
3)2 0 . 4
22. 7)0.5 8
0.3)1 1.85
0.5)3 . 4 5
8)077
Change to decimals, rounded to the nearest hundredth if the division
is uneven. Use short division.
a
b
c
d
e
23.
1 _
8 ~
1 0 _
3
7
9 —
3 _
5 ~
13. _
6 —
24.
4
7 —
1 _
4 —
14 _
5 —
15 _
8 —
3 _
8 —
141
Making and Solving Problems
For each exercise, write a question that will make a prob-
lem. Draw a ring around A. or S. or M. or D. to show
whether you will add, subtract, multiply, or divide to solve
the problem you have made. Then solve, and write the
answer.
1. Joe telephoned Bill at 10 minutes past 3 that he would
meet him in half an hour.
Question:
Solve by: A.? S.? M.? D.? Answer:
2. Kathie’s mother paid $5.96 for a set of 4 small tables.
Question:
Solve by: A.? S.? M.? D.? Answer:
3. Sylvia has saved $14.25. She wants to buy a radio that
costs $19.75.
Question:
Solve by: A.? S.? M.? D.? Answer:
4. Bill weighed 1121b. at the beginning of his vacation.
He weighed 123 lb. when he went back to school.
Question:
Solve by: A.? S.? M.? D.? Answer:
5. For the trip to the lake, Mrs. Barber bought each of
her four children a toy sailboat. The boats cost 35^ each.
Question:
Solve by: A.? S.? M.? D.? Answer: ___
6. Mrs. Knight paid $79.95 for a stove and $8 for delivery
charges. At another store, which delivers without charge,
the same stove was priced at $98.75.
Question:
Space for Work
Solve by: A.? S.? M.? D.? Answer:
142
Review of Fractions and Decimals
Add, subtract, multiply, or divide, as the signs direct. Write just
the answers on this page.
a
b
c
d
e
1.
93 1 1 _
Z4 l 2 ~
S +4i =
i +i =
6A +i =
f +2f =
2.
_ 9! _
J2 z8 ~
8| - 61 =
2§ - If =
124 - 8f =
9| - 54 =
3.
4X0.3 =
0.2 X 0.6
0.01 X 7 =
3 X 0.05 =
6 X 0.001 =
4.
3)153"
0.2]6^4
0.04)16
8)0^4
O.OOlJT
Multiply
or divide, as directed.
Show all your work.
a
b
c
5.
1 X li =
2i X If =
4f X 25 =
6.
H *5 =
3 . 9 _
8 • 16 —
5* + If =
Change to decimals. Show how you work.
7.
3 _
4 —
7 _
8 —
5 _
16 ~
8. 0.25 =
9. 14.375
0. 2|
Change to common fractions. Show each step you take.
0.375 = 0.0875 =
Round to the nearest hundredth,
b c
0.1872 2.250
1. n -T- 2 = 3J; n =
2. n = 1J X 2; n =
3. f -5- n = 1^; n =
Round to the nearest whole number.
14i
72
100
Find the value of n. (Show how you find n.)
14. 0.03 + n = 1.08; n =
15. n - 0.5 = 2.3; n =
16. 0.75 - n = 0.5; n =
3.502
24tf
143
Testing What You Have Learned
[Cumulative Review ]
Write these numbers in figures:
1. One hundred thousand six hundred fifty-seven.
2. Two million three hundred forty-six thousand.
3. Three hundred fifty and twenty-five hundredths.
4. Six and five ten- thousandths.
Find the sums.
Multiply in rows 15-18.
a
b
c
a
b
5. 3 2,189
+ 6,057
$125.75
+ 25 6.95
211,356
+ 809,798
15. 23 2
X 1 8
$1.95
X 9
c
75
X 3 6
6. 0.98 2.005
+ 1.3 +3.6
12,875
+ 9,095
16. li X 2f =
7. 18| 8. 2^
+ H + 1 6 §
17. 3J X If =
Find the remainders.
9. 15,782 $100.80 275,000
- 6,2 9 5 - 5 6.9 5 - 9,8 7 5
18. 15 4
X 0 . 0 5
28
X 1 .6
30.6
XO.O 1 5
10. 3.06
-1.125
723.4
- 548.6
2.5
-1.75
12. 12*
Hi
Divide. Write the remainders in fractions,
a b
19. 27)3 ,327 18)4 ,938
13. Write as decimals. If you cannot remem-
ber the equivalents, use short division.
a.
5. _
6 —
b.
3. _
8 —
14. Round to the nearest hundredth,
a. 2.386 c. 0.1054
b. 5.1819
144
d. 0.0071
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AR1THMET1C NE NEED
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-000031081 706-
t
education
cur. sc-ituM
559387
QA
Buswell, G. T.
106
Arithmetic we need.
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